document
stringlengths 121
3.99k
| embedding
listlengths 384
384
|
|---|---|
Edexcel AS and A level Mathematics
Pure Mathematics
Year 1 /AS
Series Editor: Harry Smith
Authors: Greg Attwood, Jack Barraclough, Ian Bettison, Alistair Macpherson, Bronwen/uni00A0Moran, Su Nicholson, Diane Oliver, Joe Petran, Keith Pledger, Harry Smith, Geoο¬ /uni00A0Staley, Robert Ward-Penny, Dave Wilkins11 β 19 PROGRESSION
|
[
-0.03336913883686066,
0.06650399416685104,
0.0637286826968193,
0.013044171966612339,
-0.09497709572315216,
-0.015243283472955227,
-0.11491992324590683,
-0.02863561548292637,
-0.0633266493678093,
-0.012000124901533127,
-0.08581973612308502,
-0.03380098193883896,
-0.03981565311551094,
0.058994095772504807,
0.004651431925594807,
-0.012597125954926014,
-0.08284824341535568,
0.020735591650009155,
-0.006416339427232742,
-0.024981413036584854,
-0.027936117723584175,
0.009922544471919537,
-0.06473630666732788,
0.015752099454402924,
0.06347917020320892,
-0.022016188129782677,
-0.009959216229617596,
0.015405949205160141,
0.0211016945540905,
-0.0795382633805275,
0.07162736356258392,
0.01790454424917698,
0.11147799342870712,
0.0001651947241043672,
-0.09481304883956909,
0.015774929895997047,
0.002477299654856324,
0.03612574189901352,
-0.028836268931627274,
-0.057243578135967255,
-0.04881604015827179,
0.013050573877990246,
-0.07012365758419037,
0.07166216522455215,
0.0773015022277832,
-0.06928139179944992,
-0.051337193697690964,
-0.04591934755444527,
-0.05236239358782768,
-0.060986388474702835,
0.0037255657371133566,
0.02911756932735443,
-0.09514743834733963,
0.06478049606084824,
-0.08300560712814331,
-0.05968974158167839,
0.05058346316218376,
0.02357746660709381,
-0.05920541286468506,
-0.06029365211725235,
-0.07886535674333572,
-0.005866901017725468,
-0.0323670320212841,
0.028328606858849525,
-0.009394900873303413,
0.023204566910862923,
0.03573627769947052,
0.08195532113313675,
0.032446105033159256,
0.05070635303854942,
-0.04393612965941429,
-0.005485734902322292,
-0.04093966633081436,
0.04306679591536522,
0.04962820187211037,
0.08311944454908371,
-0.03407031297683716,
-0.040351539850234985,
0.10759658366441727,
0.0007161606336012483,
0.0480898842215538,
0.013893643394112587,
-0.03824744001030922,
0.012447808869183064,
0.022123202681541443,
-0.02548924647271633,
0.04815473034977913,
-0.035663776099681854,
-0.04838430508971214,
0.013010960072278976,
0.12656493484973907,
-0.038143228739500046,
-0.016703980043530464,
0.03601992130279541,
0.04732649773359299,
0.015033532865345478,
-0.08883730322122574,
-0.10316774249076843,
0.08577988296747208,
0.0764169692993164,
-0.0040900008752942085,
0.06559660285711288,
-0.024193506687879562,
0.05512598156929016,
-0.09582493454217911,
-0.03492892161011696,
0.07089037448167801,
-0.08767352253198624,
0.009715401567518711,
-0.08596781641244888,
0.028402477502822876,
-0.07842099666595459,
0.02738358825445175,
-0.012850023806095123,
0.06579937040805817,
-0.04424833133816719,
0.05305372551083565,
-0.014096376486122608,
0.05621839314699173,
-0.002776580862700939,
-0.017707645893096924,
0.005692173261195421,
0.018446218222379684,
0.09253835678100586,
0.005531358532607555,
0.011495662853121758,
-0.007608858868479729,
0.07724502682685852,
-0.04045984521508217,
-0.016140103340148926,
0.054105229675769806,
0.03115791454911232,
0.042396701872348785,
0.016307570040225983,
-0.05268661305308342,
-0.03897087648510933,
0.05631459876894951,
0.033776018768548965,
-0.019315635785460472,
0.09391386061906815,
0.06577903777360916,
0.04821992293000221,
-0.025968989357352257,
-0.041904617100954056,
-0.07086727023124695,
-0.020724808797240257,
0.02417961321771145,
0.02586485631763935,
-0.001228907029144466,
0.038699302822351456,
0.01962890662252903,
-0.07445389032363892,
-0.012460963800549507,
0.047594182193279266,
-0.00394991971552372,
0.029909642413258553,
0.07180360704660416,
-0.004741004668176174,
0.009283088147640228,
-0.0022170059382915497,
-0.1225944459438324,
-0.04187113419175148,
-0.04533099755644798,
0.08706812560558319,
0.05563156306743622,
-0.12001992017030716,
0.014721821993589401,
0.003754799487069249,
-0.11474263668060303,
0.00027262355433776975,
0.009269568137824535,
-0.0334496796131134,
-0.06735676527023315,
0.022616812959313393,
-0.01609640009701252,
-0.01849399320781231,
0.09989286214113235,
0.056767553091049194,
0.006745698396116495,
-0.042928338050842285,
-0.024936528876423836,
-0.08883431553840637,
0.015168909914791584,
-0.051316019147634506,
0.014170289970934391,
0.0363365113735199,
-0.02239842526614666,
0.06280101090669632,
-0.0065349978394806385,
-0.0018926695920526981,
-0.03278503194451332,
0.009898318909108639,
-0.08881789445877075,
-0.02264726720750332,
-0.05511254817247391,
0.0264622513204813,
0.04229371249675751,
-0.004673129878938198,
-0.04245034605264664,
0.05768512189388275,
-0.029551250860095024,
0.048099543899297714,
0.056169543415308,
-0.018705978989601135,
0.05040409415960312,
-0.15582074224948883,
0.0911090224981308,
-0.09640508890151978,
-0.01805403269827366,
0.021169880405068398,
-0.05102244392037392,
-0.08560652285814285,
-0.007924223318696022,
-0.07149598747491837,
-0.02433682791888714,
-0.012348931282758713,
0.022485313937067986,
-0.014772942289710045,
-0.04600349813699722,
-0.03183776140213013,
0.04853089898824692,
0.05318666622042656,
0.06850938498973846,
-2.7863769318387727e-33,
-0.05947592854499817,
0.0025620257947593927,
-0.08635429292917252,
-0.001822443911805749,
0.01782630942761898,
0.005243205931037664,
-0.0378694087266922,
-0.008002913556993008,
0.09634296596050262,
0.029561810195446014,
0.01674458011984825,
0.054908350110054016,
0.03465265780687332,
0.0412580668926239,
-0.07801186293363571,
-0.08578294515609741,
-0.1501738727092743,
0.0422632098197937,
-0.01593286357820034,
-0.08267267048358917,
-0.0690629854798317,
0.023932093754410744,
-0.030415935441851616,
-0.02723657712340355,
-0.0016115965554490685,
0.02030026540160179,
0.016589578241109848,
0.07667302340269089,
-0.06325361132621765,
0.07069548219442368,
-0.001289432868361473,
-0.06715758889913559,
0.000836844788864255,
0.04479791969060898,
-0.08537901937961578,
0.03230264037847519,
0.0791788324713707,
-0.07539331167936325,
-0.062108974903821945,
-0.011865988373756409,
0.04193475469946861,
-0.042574960738420486,
0.024772154167294502,
-0.05757302790880203,
0.05301082134246826,
0.06803672760725021,
-0.017581569030880928,
0.08955132216215134,
-0.027654634788632393,
0.007700539659708738,
-0.10239279270172119,
0.09694798290729523,
-0.003147023729979992,
-0.010930005460977554,
0.02420753985643387,
0.02955188974738121,
-0.039429277181625366,
-0.011724425479769707,
0.009134535677731037,
-0.009330174885690212,
-0.05215127021074295,
-0.028863538056612015,
0.0467342808842659,
0.04070490226149559,
0.08582364022731781,
0.03476772457361221,
-0.047006528824567795,
-0.0429961197078228,
-0.002980724908411503,
-0.01573079079389572,
0.013077553361654282,
-0.003517911769449711,
-0.11049264669418335,
-0.1319512575864792,
-0.021319178864359856,
-0.03293623775243759,
0.024244725704193115,
0.005064776167273521,
-0.044493723660707474,
0.010805854573845863,
-0.03786711022257805,
-0.029493654146790504,
0.04007545858621597,
0.0810803696513176,
0.04084475338459015,
0.006699732970446348,
0.09047569334506989,
0.007945993915200233,
-0.06090029329061508,
0.06816446781158447,
0.03028530813753605,
-0.06428275257349014,
-0.014916921965777874,
-0.004809353966265917,
0.020346783101558685,
9.811003227139516e-32,
-0.04323093965649605,
0.031207680702209473,
-0.08056933432817459,
-0.00008944700675783679,
0.06656002253293991,
0.07017149031162262,
0.036527104675769806,
-0.0334068164229393,
0.03151514753699303,
0.007334255613386631,
0.04442734271287918,
-0.035288430750370026,
-0.0318467952311039,
0.007563494145870209,
0.01900760643184185,
0.04923078790307045,
-0.04111557826399803,
0.01100613921880722,
-0.00662855664268136,
0.027157891541719437,
0.011209872551262379,
0.05485932156443596,
-0.029557034373283386,
-0.0044672125950455666,
-0.023313313722610474,
0.027583857998251915,
-0.020261133089661598,
-0.025812407955527306,
-0.011933678761124611,
0.03216150775551796,
-0.0024420435074716806,
-0.006210526451468468,
0.023102538660168648,
-0.02152712643146515,
-0.01680789142847061,
0.07351034879684448,
0.03799882158637047,
0.028039593249559402,
-0.03587900102138519,
-0.026390204206109047,
-0.019808193668723106,
-0.1087745800614357,
0.06377632915973663,
-0.015307330526411533,
0.07312050461769104,
0.02818765863776207,
-0.003689306555315852,
0.04361303150653839,
0.02603779546916485,
-0.024831781163811684,
-0.02062988467514515,
-0.025229601189494133,
-0.013872956857085228,
0.004171317908912897,
0.07868484407663345,
-0.006786867044866085,
-0.04533401131629944,
0.011305999010801315,
-0.1127791628241539,
0.05228410288691521,
0.09482375532388687,
0.03608499467372894,
-0.047397494316101074,
0.013440298847854137
] |
Edexcel AS and A level Mathematics
Pure Mathematics
Year 1 /AS
Series Editor: Harry Smith
Authors: Greg Attwood, Jack Barraclough, Ian Bettison, Alistair Macpherson, Bronwen/uni00A0Moran, Su Nicholson, Diane Oliver, Joe Petran, Keith Pledger, Harry Smith, Geoο¬ /uni00A0Staley, Robert Ward-Penny, Dave Wilkins11 β 19 PROGRESSION
|
[
-0.03336913883686066,
0.06650399416685104,
0.0637286826968193,
0.013044171966612339,
-0.09497709572315216,
-0.015243283472955227,
-0.11491992324590683,
-0.02863561548292637,
-0.0633266493678093,
-0.012000124901533127,
-0.08581973612308502,
-0.03380098193883896,
-0.03981565311551094,
0.058994095772504807,
0.004651431925594807,
-0.012597125954926014,
-0.08284824341535568,
0.020735591650009155,
-0.006416339427232742,
-0.024981413036584854,
-0.027936117723584175,
0.009922544471919537,
-0.06473630666732788,
0.015752099454402924,
0.06347917020320892,
-0.022016188129782677,
-0.009959216229617596,
0.015405949205160141,
0.0211016945540905,
-0.0795382633805275,
0.07162736356258392,
0.01790454424917698,
0.11147799342870712,
0.0001651947241043672,
-0.09481304883956909,
0.015774929895997047,
0.002477299654856324,
0.03612574189901352,
-0.028836268931627274,
-0.057243578135967255,
-0.04881604015827179,
0.013050573877990246,
-0.07012365758419037,
0.07166216522455215,
0.0773015022277832,
-0.06928139179944992,
-0.051337193697690964,
-0.04591934755444527,
-0.05236239358782768,
-0.060986388474702835,
0.0037255657371133566,
0.02911756932735443,
-0.09514743834733963,
0.06478049606084824,
-0.08300560712814331,
-0.05968974158167839,
0.05058346316218376,
0.02357746660709381,
-0.05920541286468506,
-0.06029365211725235,
-0.07886535674333572,
-0.005866901017725468,
-0.0323670320212841,
0.028328606858849525,
-0.009394900873303413,
0.023204566910862923,
0.03573627769947052,
0.08195532113313675,
0.032446105033159256,
0.05070635303854942,
-0.04393612965941429,
-0.005485734902322292,
-0.04093966633081436,
0.04306679591536522,
0.04962820187211037,
0.08311944454908371,
-0.03407031297683716,
-0.040351539850234985,
0.10759658366441727,
0.0007161606336012483,
0.0480898842215538,
0.013893643394112587,
-0.03824744001030922,
0.012447808869183064,
0.022123202681541443,
-0.02548924647271633,
0.04815473034977913,
-0.035663776099681854,
-0.04838430508971214,
0.013010960072278976,
0.12656493484973907,
-0.038143228739500046,
-0.016703980043530464,
0.03601992130279541,
0.04732649773359299,
0.015033532865345478,
-0.08883730322122574,
-0.10316774249076843,
0.08577988296747208,
0.0764169692993164,
-0.0040900008752942085,
0.06559660285711288,
-0.024193506687879562,
0.05512598156929016,
-0.09582493454217911,
-0.03492892161011696,
0.07089037448167801,
-0.08767352253198624,
0.009715401567518711,
-0.08596781641244888,
0.028402477502822876,
-0.07842099666595459,
0.02738358825445175,
-0.012850023806095123,
0.06579937040805817,
-0.04424833133816719,
0.05305372551083565,
-0.014096376486122608,
0.05621839314699173,
-0.002776580862700939,
-0.017707645893096924,
0.005692173261195421,
0.018446218222379684,
0.09253835678100586,
0.005531358532607555,
0.011495662853121758,
-0.007608858868479729,
0.07724502682685852,
-0.04045984521508217,
-0.016140103340148926,
0.054105229675769806,
0.03115791454911232,
0.042396701872348785,
0.016307570040225983,
-0.05268661305308342,
-0.03897087648510933,
0.05631459876894951,
0.033776018768548965,
-0.019315635785460472,
0.09391386061906815,
0.06577903777360916,
0.04821992293000221,
-0.025968989357352257,
-0.041904617100954056,
-0.07086727023124695,
-0.020724808797240257,
0.02417961321771145,
0.02586485631763935,
-0.001228907029144466,
0.038699302822351456,
0.01962890662252903,
-0.07445389032363892,
-0.012460963800549507,
0.047594182193279266,
-0.00394991971552372,
0.029909642413258553,
0.07180360704660416,
-0.004741004668176174,
0.009283088147640228,
-0.0022170059382915497,
-0.1225944459438324,
-0.04187113419175148,
-0.04533099755644798,
0.08706812560558319,
0.05563156306743622,
-0.12001992017030716,
0.014721821993589401,
0.003754799487069249,
-0.11474263668060303,
0.00027262355433776975,
0.009269568137824535,
-0.0334496796131134,
-0.06735676527023315,
0.022616812959313393,
-0.01609640009701252,
-0.01849399320781231,
0.09989286214113235,
0.056767553091049194,
0.006745698396116495,
-0.042928338050842285,
-0.024936528876423836,
-0.08883431553840637,
0.015168909914791584,
-0.051316019147634506,
0.014170289970934391,
0.0363365113735199,
-0.02239842526614666,
0.06280101090669632,
-0.0065349978394806385,
-0.0018926695920526981,
-0.03278503194451332,
0.009898318909108639,
-0.08881789445877075,
-0.02264726720750332,
-0.05511254817247391,
0.0264622513204813,
0.04229371249675751,
-0.004673129878938198,
-0.04245034605264664,
0.05768512189388275,
-0.029551250860095024,
0.048099543899297714,
0.056169543415308,
-0.018705978989601135,
0.05040409415960312,
-0.15582074224948883,
0.0911090224981308,
-0.09640508890151978,
-0.01805403269827366,
0.021169880405068398,
-0.05102244392037392,
-0.08560652285814285,
-0.007924223318696022,
-0.07149598747491837,
-0.02433682791888714,
-0.012348931282758713,
0.022485313937067986,
-0.014772942289710045,
-0.04600349813699722,
-0.03183776140213013,
0.04853089898824692,
0.05318666622042656,
0.06850938498973846,
-2.7863769318387727e-33,
-0.05947592854499817,
0.0025620257947593927,
-0.08635429292917252,
-0.001822443911805749,
0.01782630942761898,
0.005243205931037664,
-0.0378694087266922,
-0.008002913556993008,
0.09634296596050262,
0.029561810195446014,
0.01674458011984825,
0.054908350110054016,
0.03465265780687332,
0.0412580668926239,
-0.07801186293363571,
-0.08578294515609741,
-0.1501738727092743,
0.0422632098197937,
-0.01593286357820034,
-0.08267267048358917,
-0.0690629854798317,
0.023932093754410744,
-0.030415935441851616,
-0.02723657712340355,
-0.0016115965554490685,
0.02030026540160179,
0.016589578241109848,
0.07667302340269089,
-0.06325361132621765,
0.07069548219442368,
-0.001289432868361473,
-0.06715758889913559,
0.000836844788864255,
0.04479791969060898,
-0.08537901937961578,
0.03230264037847519,
0.0791788324713707,
-0.07539331167936325,
-0.062108974903821945,
-0.011865988373756409,
0.04193475469946861,
-0.042574960738420486,
0.024772154167294502,
-0.05757302790880203,
0.05301082134246826,
0.06803672760725021,
-0.017581569030880928,
0.08955132216215134,
-0.027654634788632393,
0.007700539659708738,
-0.10239279270172119,
0.09694798290729523,
-0.003147023729979992,
-0.010930005460977554,
0.02420753985643387,
0.02955188974738121,
-0.039429277181625366,
-0.011724425479769707,
0.009134535677731037,
-0.009330174885690212,
-0.05215127021074295,
-0.028863538056612015,
0.0467342808842659,
0.04070490226149559,
0.08582364022731781,
0.03476772457361221,
-0.047006528824567795,
-0.0429961197078228,
-0.002980724908411503,
-0.01573079079389572,
0.013077553361654282,
-0.003517911769449711,
-0.11049264669418335,
-0.1319512575864792,
-0.021319178864359856,
-0.03293623775243759,
0.024244725704193115,
0.005064776167273521,
-0.044493723660707474,
0.010805854573845863,
-0.03786711022257805,
-0.029493654146790504,
0.04007545858621597,
0.0810803696513176,
0.04084475338459015,
0.006699732970446348,
0.09047569334506989,
0.007945993915200233,
-0.06090029329061508,
0.06816446781158447,
0.03028530813753605,
-0.06428275257349014,
-0.014916921965777874,
-0.004809353966265917,
0.020346783101558685,
9.811003227139516e-32,
-0.04323093965649605,
0.031207680702209473,
-0.08056933432817459,
-0.00008944700675783679,
0.06656002253293991,
0.07017149031162262,
0.036527104675769806,
-0.0334068164229393,
0.03151514753699303,
0.007334255613386631,
0.04442734271287918,
-0.035288430750370026,
-0.0318467952311039,
0.007563494145870209,
0.01900760643184185,
0.04923078790307045,
-0.04111557826399803,
0.01100613921880722,
-0.00662855664268136,
0.027157891541719437,
0.011209872551262379,
0.05485932156443596,
-0.029557034373283386,
-0.0044672125950455666,
-0.023313313722610474,
0.027583857998251915,
-0.020261133089661598,
-0.025812407955527306,
-0.011933678761124611,
0.03216150775551796,
-0.0024420435074716806,
-0.006210526451468468,
0.023102538660168648,
-0.02152712643146515,
-0.01680789142847061,
0.07351034879684448,
0.03799882158637047,
0.028039593249559402,
-0.03587900102138519,
-0.026390204206109047,
-0.019808193668723106,
-0.1087745800614357,
0.06377632915973663,
-0.015307330526411533,
0.07312050461769104,
0.02818765863776207,
-0.003689306555315852,
0.04361303150653839,
0.02603779546916485,
-0.024831781163811684,
-0.02062988467514515,
-0.025229601189494133,
-0.013872956857085228,
0.004171317908912897,
0.07868484407663345,
-0.006786867044866085,
-0.04533401131629944,
0.011305999010801315,
-0.1127791628241539,
0.05228410288691521,
0.09482375532388687,
0.03608499467372894,
-0.047397494316101074,
0.013440298847854137
] |
iiContents
Overarching themes iv
Extra online c
ontent vi
1 Algebraic e
xpressions 1
1.1 Index law
s 2
1.2 Expanding brack
ets 4
1.3 Factorising 6
1.4 Negative and fractional indic
es 9
1.5 Surds 12
1.6 Rationalising denominators 13
Mixed ex
ercise 1 15
2 Quadratics 18
2.1 Solving quadratic equations 19
2.2 Completing the squar
e 22
2.3 Functions 25
2.4 Quadratic graphs 27
2.5 The discriminant 30
2.6 Modelling with quadratics 32
Mixed ex
ercise 2 35
3 Equations and inequalities 38
3.1 Linear simultaneous equations 39
3.2 Quadratic simultaneous equations 41
3.3 Simultaneous equations on graphs 42
3.4 Linear inequalities 46
3.5 Quadratic inequalities 48
3.6 Inequalities on graphs 51
3.7 Regions 53
Mixed ex
ercise 3 56
4 Graphs and trans
formations 59
4.1 Cubic graphs 60
4.2 Quartic graphs 64
4.3 Reciprocal gr
aphs 66
4.4 Points of int
ersection 68
4.5 Translating gr
aphs 71 Contents
4.6 Stretching graphs 75
4.7 Trans
forming functions 79
Mixed ex
ercise 4 82
Review ex
ercise 1 85
5 Straight line gr
aphs 89
5.1 y = mx
+ c 90
5.2 Equations of st
raight lines 93
5.3 Parall
el and perpendicular lines 97
5.4 Length and area 100
5.5 Modelling with straight lines 103
Mixed ex
ercise 5 108
6 Circles 113
6.1 Midpoints and perpendicular
bisectors 114
6.2 Equation of a cir
cle 117
6.3 Intersections of st
raight lines
and circles 121
6.4 Use tangent and chord pr
operties 123
6.5 Circles and t
riangles 128
Mixed ex
ercise 6 132
7 Algebraic methods 137
7.1 Algebraic fr
actions 138
7.2 Dividing polynomials 139
7.3 The factor theorem 143
7.4 Mathematical proof 146
7.5 Methods of proo
f 150
Mixed ex
ercise 7 154
8 The binomial expansion 158
8.1 Pascal
βs triangle 159
8.2 Factorial notation 161
8.3 The binomial expansion 163
8.4 Solving binomial problems 165
|
[
-0.04587770253419876,
0.05886335298418999,
0.004491395317018032,
-0.0328848622739315,
0.05864277854561806,
-0.07106245309114456,
-0.051982954144477844,
0.06517630815505981,
-0.13637033104896545,
0.021316474303603172,
0.03795776143670082,
-0.04442531242966652,
0.030444122850894928,
0.01262225303798914,
-0.04015053063631058,
0.021946460008621216,
-0.01325657032430172,
0.08812645077705383,
-0.030483566224575043,
-0.06723654270172119,
0.007445794530212879,
-0.011411849409341812,
-0.07661040872335434,
-0.01756015047430992,
0.05649947002530098,
-0.04842269420623779,
0.034893304109573364,
0.0609990730881691,
0.01524603646248579,
-0.07718601077795029,
0.024323198944330215,
0.0777951031923294,
0.15447360277175903,
0.03681764379143715,
-0.04492190480232239,
-0.05514458939433098,
0.0992051437497139,
0.04703432694077492,
-0.03154910355806351,
0.02373882383108139,
-0.03896920382976532,
0.01328359730541706,
-0.027178671211004257,
0.05244726315140724,
0.07780677825212479,
-0.07876717299222946,
-0.021858487278223038,
-0.03963248431682587,
-0.09094985574483871,
-0.015994055196642876,
-0.0046511017717421055,
-0.03183234855532646,
-0.11674840748310089,
0.047885797917842865,
-0.030176814645528793,
-0.03135516494512558,
-0.03832686319947243,
-0.0189603753387928,
-0.0662032961845398,
0.020540839061141014,
0.026918988674879074,
-0.02038589119911194,
0.02901977300643921,
0.07430344820022583,
0.009440682828426361,
0.029622400179505348,
0.08452539891004562,
0.022804751992225647,
-0.016427429392933846,
0.10833840072154999,
-0.050889551639556885,
0.010864319279789925,
-0.07875269651412964,
0.072982057929039,
0.03928345441818237,
-0.030413605272769928,
-0.050683289766311646,
-0.014454636722803116,
-0.050190944224596024,
-0.060023996978998184,
-0.016449088230729103,
-0.037618156522512436,
0.059137992560863495,
0.017451681196689606,
-0.012264491058886051,
-0.055149007588624954,
0.08387143164873123,
0.021706221625208855,
0.035885248333215714,
0.01684240810573101,
0.012448583729565144,
-0.008506480604410172,
0.023457488045096397,
-0.09311680495738983,
-0.03130929544568062,
-0.004092910327017307,
-0.01789214089512825,
-0.11796947568655014,
0.04524734988808632,
0.11740800738334656,
0.008967871777713299,
-0.023265782743692398,
0.050371814519166946,
0.013689511455595493,
-0.03896902874112129,
-0.07775389403104782,
-0.0058567277155816555,
-0.04160628467798233,
0.05004856735467911,
-0.0680709257721901,
-0.09325707703828812,
-0.0005547126056626439,
-0.023552455008029938,
-0.028807472437620163,
0.03855929896235466,
-0.11231481283903122,
0.12572991847991943,
-0.018831094726920128,
0.08288327604532242,
-0.014220157638192177,
0.043577637523412704,
0.024865565821528435,
0.03739510476589203,
-0.012909431010484695,
0.04437506943941116,
-0.04540128633379936,
-0.07363544404506683,
0.02759787067770958,
-0.02665005251765251,
-0.02346339076757431,
0.049966663122177124,
0.05197341740131378,
0.03148600831627846,
-0.030340218916535378,
-0.05616261437535286,
-0.08669774979352951,
-0.008926409296691418,
0.13895180821418762,
0.03455169126391411,
0.06186820939183235,
0.013875891454517841,
0.12260972708463669,
0.09350989758968353,
-0.026297632604837418,
-0.11900275200605392,
-0.025693640112876892,
-0.02081782929599285,
-0.026180991902947426,
0.05069952830672264,
-0.0024244196247309446,
-0.006102369166910648,
-0.0635986477136612,
0.04623527079820633,
0.056289128959178925,
0.03634559363126755,
0.010417974554002285,
0.07773242145776749,
-0.04517738148570061,
-0.0002246892254333943,
0.04896790161728859,
-0.038842152804136276,
0.003629087470471859,
-0.012868908233940601,
0.10930848121643066,
-0.015662075951695442,
-0.11160644888877869,
0.019765306264162064,
0.05277584493160248,
-0.03308830037713051,
0.003441514912992716,
0.060153499245643616,
-0.007457762025296688,
0.01295776292681694,
0.07185829430818558,
0.009448571130633354,
-0.01940307579934597,
0.06432772427797318,
0.012641966342926025,
-0.02101978100836277,
0.01371054444462061,
0.0763704851269722,
-0.02430162951350212,
-0.03044079802930355,
0.058981090784072876,
-0.005359061062335968,
-0.018593156710267067,
0.009558470919728279,
0.04228128120303154,
0.01842414028942585,
-0.08866041898727417,
-0.017139866948127747,
-0.05591066926717758,
-0.04490770399570465,
0.07045283913612366,
-0.07752528786659241,
-0.11588422954082489,
0.03315313532948494,
0.015297284349799156,
-0.09677330404520035,
0.1032436266541481,
-0.030106665566563606,
0.003120665205642581,
0.04696133732795715,
0.018057404085993767,
-0.014956718310713768,
-0.019840002059936523,
-0.06132609769701958,
-0.022755200043320656,
-0.04644618183374405,
0.05882929265499115,
-0.008376491256058216,
-0.04797128215432167,
-0.017019221559166908,
-0.0906599685549736,
0.03790006414055824,
-0.00620553782209754,
0.02114819549024105,
-0.028709448873996735,
-0.08228772133588791,
-0.03760671988129616,
0.0416107103228569,
-0.03373990207910538,
0.06056085601449013,
2.6918479740468645e-32,
-0.05241548269987106,
-0.0886208638548851,
-0.1177949607372284,
0.07375747710466385,
-0.007697864901274443,
-0.039868924766778946,
0.02881450392305851,
-0.004682576283812523,
0.04997146502137184,
0.0007359904702752829,
0.036829810589551926,
0.02715986594557762,
0.01779339462518692,
0.022677315399050713,
-0.05188966542482376,
-0.0005349344573915005,
-0.006524225231260061,
0.03727421537041664,
0.015898065641522408,
0.0728144645690918,
0.007793158758431673,
0.0817110538482666,
0.015677694231271744,
-0.015599258244037628,
-0.035638414323329926,
0.08891259133815765,
-0.040773674845695496,
-0.10394738614559174,
-0.07934942096471786,
0.0649714544415474,
-0.019749116152524948,
-0.021956585347652435,
-0.020536448806524277,
0.03251578286290169,
-0.04854551702737808,
-0.02307622693479061,
0.010874629952013493,
-0.043210215866565704,
-0.013128921389579773,
0.05914819985628128,
0.04948816075921059,
0.027550475671887398,
0.07367950677871704,
0.0005530361668206751,
0.07820110768079758,
-0.04679021239280701,
0.05194203928112984,
0.02290436625480652,
-0.012530382722616196,
0.013470933772623539,
-0.06579434871673584,
0.018798183649778366,
0.019559072330594063,
0.0371769480407238,
0.02559063397347927,
-0.02393052726984024,
0.04107338935136795,
-0.02699189819395542,
-0.0032611265778541565,
0.03284269943833351,
-0.05111970379948616,
-0.03262863680720329,
-0.04772798344492912,
0.08769656717777252,
-0.012116145342588425,
-0.067836232483387,
-0.03737970069050789,
0.006317612715065479,
0.004008420743048191,
0.03709280863404274,
0.026202980428934097,
0.053672563284635544,
-0.05520159751176834,
-0.06117885932326317,
-0.07944245636463165,
-0.029750382527709007,
0.062164388597011566,
0.05473023280501366,
0.004281812347471714,
-0.021050503477454185,
-0.09914233535528183,
0.02442150004208088,
0.02090461738407612,
-0.052362021058797836,
0.0009358221432194114,
-0.0040595512837171555,
0.04046294093132019,
0.07271140813827515,
-0.00988854467868805,
-0.01859181374311447,
-0.019837044179439545,
-0.01557493768632412,
-0.020160431042313576,
0.007296545431017876,
0.00579999340698123,
7.164122613989007e-32,
-0.011047374457120895,
0.021943341940641403,
-0.05114409700036049,
0.07265076041221619,
-0.000044346092181513086,
-0.0008732072310522199,
0.04799771308898926,
-0.013833470642566681,
0.009202607907354832,
-0.02177286334335804,
0.10692160576581955,
0.04912129417061806,
0.038110096007585526,
0.056879155337810516,
-0.03284703567624092,
0.01888175494968891,
0.014863033778965473,
0.10711297392845154,
-0.06260593980550766,
0.01953052543103695,
-0.021296370774507523,
0.06788598001003265,
-0.0410517081618309,
-0.030784009024500847,
0.03163784369826317,
0.04760054871439934,
-0.015593339689075947,
-0.015329576097428799,
0.02588268555700779,
-0.03054201602935791,
0.04254258796572685,
0.05768487602472305,
-0.041294779628515244,
-0.06933314353227615,
-0.02101362869143486,
0.0863417237997055,
-0.020082611590623856,
0.08621503412723541,
-0.05466308817267418,
0.025188008323311806,
-0.027070241048932076,
-0.07519817352294922,
0.021219713613390923,
0.044465936720371246,
0.04311688616871834,
-0.043654296547174454,
0.03376909717917442,
-0.03319994732737541,
0.06653057038784027,
0.022079454734921455,
-0.05758308246731758,
-0.05058729648590088,
-0.072236567735672,
0.0013795208651572466,
0.08442208170890808,
-0.05746118351817131,
-0.05684658885002136,
0.048944223672151566,
0.025690382346510887,
-0.026998501271009445,
0.043778836727142334,
0.07759258896112442,
-0.06801765412092209,
-0.01995903067290783
] |
iiiContents
8.5 Binomial estimation 167
Mixed ex
ercise 8 169
9 Trigonometric r
atios 173
9.1 The cosine rul
e 174
9.2 The sine rule 179
9.3 Areas o
f triangles 185
9.4 Solving triangle pr
oblems 187
9.5 Graphs of sine, c
osine and tangent 192
9.6 Trans
forming trigonometric graphs 194
Mixed ex
ercise 9 198
10 Trigonometric identities and
equations 202
10.1 Angles in all four quadr
ants 203
10.2 Exact values o
f trigonometrical ratios 208
10.3 Trigonomet
ric identities 209
10.4 Simple trig
onometric equations 213
10.5 Harder trig
onometric equations 217
10.6 Equations and identities 219
Mixed ex
ercise 10 222
Review ex
ercise 2 226
11 Vectors 230
11.1 Vectors 231
11.2 Representing v
ectors 235
11.3 Magnitude and direction 239
11.4 Position v
ectors 242
11.5 Solving geometric pr
oblems 244
11.6 Modelling with vectors 248
Mixed ex
ercise 11 251
12 Differentiation 255
12.1 Gradients of cur
ves 256
12.2 Finding the derivative 259
12.3 Differentiating
xn 262
12.4 Differentiating quadr
atics 26412.5 Differentiating functions with t
wo
or more terms 266
12.6 Gradients, tang
ents and normal 268
12.7 Increasing and decr
easing functions 270
12.8 Second order deriv
atives 271
12.9 Stationary points 273
12.10 Sketching gr
adient functions 277
12.11 Modelling with differentiation 279
Mixed ex
ercise 12 282
13 Integration 287
13.1 Integr
ating xn 288
13.2 Indefinite integr
als 290
13.3 Finding functions 293
13.4 Definite integr
als 295
13.5 Areas under cur
ves 297
13.6 Areas under the
x-axis 300
13.7 Areas bet
ween curves and lines 302
Mixed ex
ercise 13 306
14 Exponentials and logarithms 311
14.1 Exponential functions 312
14.2 y = ex 314
14.3 Exponential modelling 317
14.4 Logarithms 319
14.5 Law
s of logarithms 321
14.6 Solving equations using logarithms 324
14.7 Working with natur
al logarithms 326
14.8 Logarithms and non-linear data 328
Mixed ex
ercise 14 334
Review ex
ercise 3 338
Practic
e exam paper 342
Answ
ers 345
Index 399
|
[
-0.008981455117464066,
0.044350121170282364,
-0.0005638255388475955,
0.006036241538822651,
-0.04708579555153847,
-0.04335619509220123,
-0.023125313222408295,
0.0037964177317917347,
-0.0755218043923378,
-0.02962755225598812,
0.04980447143316269,
-0.05688834190368652,
-0.04616783186793327,
0.059873055666685104,
0.04461919888854027,
0.03873879462480545,
-0.016911735758185387,
0.03095100075006485,
0.004605743568390608,
-0.0907306969165802,
-0.018681248649954796,
-0.033709701150655746,
0.03735131397843361,
-0.08571986854076385,
0.015195919200778008,
0.05045587942004204,
0.011527260765433311,
0.034753598272800446,
-0.0013671745546162128,
0.018671289086341858,
0.015959693118929863,
0.06733540445566177,
0.06274273246526718,
-0.07206027954816818,
-0.02000609040260315,
-0.09163984656333923,
-0.026250528171658516,
0.013739614747464657,
0.04097476974129677,
0.07923601567745209,
-0.02144515886902809,
0.06431929022073746,
0.007131035439670086,
0.056296925991773605,
0.005879227537661791,
-0.007880364544689655,
-0.04260189086198807,
-0.013974917121231556,
0.0041568707674741745,
0.06024773791432381,
0.03679313138127327,
0.049578309059143066,
-0.17063315212726593,
-0.02479337900876999,
0.027898892760276794,
-0.012131843715906143,
-0.0465543270111084,
-0.05627572163939476,
-0.09891510754823685,
-0.05114027485251427,
0.022198200225830078,
0.01583203300833702,
-0.000030200490073184483,
0.026346351951360703,
0.002391509246081114,
0.04678938165307045,
0.03300276771187782,
-0.06258876621723175,
0.010984901338815689,
0.03022921085357666,
-0.08735182136297226,
0.05333821475505829,
-0.037486396729946136,
0.020341716706752777,
-0.02687332034111023,
0.0014027805300429463,
-0.04860685393214226,
0.036256615072488785,
-0.05245377868413925,
-0.09031933546066284,
-0.12091473489999771,
0.03678543493151665,
0.023899156600236893,
-0.02562960796058178,
0.04966242238879204,
0.08379779756069183,
0.07199225574731827,
0.17676153779029846,
0.04368804767727852,
-0.014867797493934631,
0.03769073262810707,
-0.06662698090076447,
-0.04276487976312637,
-0.016017049551010132,
-0.002096223644912243,
0.09903940558433533,
-0.006674441043287516,
-0.0926181897521019,
-0.0019107984844595194,
0.11494167894124985,
-0.02601550705730915,
-0.030301792547106743,
0.029622208327054977,
0.030997274443507195,
-0.07819755375385284,
0.01989930123090744,
0.08006196469068527,
-0.03367527946829796,
0.030994590371847153,
-0.039485521614551544,
-0.09987173974514008,
0.036876413971185684,
0.05349583923816681,
0.03246116265654564,
-0.03426169604063034,
-0.07439500838518143,
0.09857427328824997,
0.02705060876905918,
0.09755557775497437,
0.06048297509551048,
0.007533743511885405,
-0.04268748313188553,
0.010382222943007946,
0.048348166048526764,
0.01190288458019495,
-0.051227957010269165,
-0.05843409523367882,
-0.032567817717790604,
-0.06378181278705597,
0.054090406745672226,
0.08441084623336792,
0.015322351828217506,
0.0179911982268095,
-0.059803180396556854,
-0.059874214231967926,
-0.047950949519872665,
0.07749722898006439,
0.11264655739068985,
-0.013847914524376392,
0.03878270834684372,
-0.00020463604596443474,
0.04831729829311371,
0.060104724019765854,
0.019569573923945427,
-0.03158031776547432,
-0.03263121470808983,
-0.08875005692243576,
0.032659661024808884,
-0.04263477399945259,
0.038515836000442505,
0.040568143129348755,
0.05920872464776039,
0.005286280997097492,
0.11335717141628265,
-0.02263355627655983,
0.07702198624610901,
0.008570096455514431,
-0.02224496752023697,
0.05501497536897659,
-0.005590035114437342,
-0.005793755874037743,
0.033288732171058655,
0.04391296207904816,
0.060676004737615585,
0.01128992810845375,
-0.03313298523426056,
0.039029184728860855,
0.012984024360775948,
0.0008200151496566832,
-0.02144072949886322,
0.04562520235776901,
0.02444368042051792,
-0.0024520456790924072,
0.01397097297012806,
-0.004146779887378216,
0.001262432779185474,
0.1306314766407013,
0.04450279101729393,
0.009163297712802887,
0.01543730590492487,
-0.004273469094187021,
-0.031110314652323723,
-0.007516044192016125,
0.040987465530633926,
-0.016445200890302658,
-0.04013670235872269,
-0.06844396144151688,
0.05225982144474983,
-0.010169723071157932,
-0.01320390123873949,
0.042253490537405014,
-0.03912078216671944,
-0.02322397194802761,
0.014752950519323349,
-0.09438613802194595,
-0.07442621886730194,
0.026991019025444984,
-0.01782792992889881,
-0.05069855600595474,
0.047796186059713364,
0.043119680136442184,
-0.05021870881319046,
0.009241556748747826,
-0.0007403453928418458,
0.033376678824424744,
0.020678305998444557,
0.031430285423994064,
-0.07802574336528778,
-0.05331725254654884,
0.00376264750957489,
-0.02354312688112259,
-0.14156083762645721,
0.0005965960444882512,
-0.03159293532371521,
-0.010823395103216171,
-0.042197488248348236,
0.013301251456141472,
-0.06163651496171951,
-0.04460570588707924,
0.02200264111161232,
-0.02163686975836754,
-0.08430630713701248,
-0.00840936228632927,
1.310664887033724e-32,
-0.11692920327186584,
-0.030584905296564102,
-0.09121321886777878,
-0.0204591266810894,
0.004125443287193775,
-0.022564545273780823,
0.11625669151544571,
0.0009431121870875359,
0.08987400680780411,
-0.004569002892822027,
0.06712193042039871,
0.03477011248469353,
-0.006768278311938047,
-0.09341101348400116,
0.018054110929369926,
-0.007524969056248665,
0.003951126243919134,
0.0012845287565141916,
-0.01162766944617033,
-0.03876559063792229,
-0.0028441143222153187,
0.08471200615167618,
0.06878334283828735,
0.014121274463832378,
-0.0465356707572937,
0.042327359318733215,
0.05020098388195038,
-0.11798606812953949,
-0.1501525640487671,
-0.03199579194188118,
-0.007299718447029591,
-0.007269751280546188,
0.05657526105642319,
0.05427397042512894,
-0.04557511955499649,
-0.09654051810503006,
0.04103083536028862,
0.025986984372138977,
-0.03970777615904808,
-0.06744148582220078,
0.033260367810726166,
0.002082968130707741,
0.042895544320344925,
0.011189596727490425,
0.024091502651572227,
-0.005623709876090288,
0.0035137978848069906,
-0.040970996022224426,
-0.03340372070670128,
0.004018967505544424,
-0.013160218484699726,
-0.03128720074892044,
-0.011295359581708908,
-0.018929999321699142,
0.09278006106615067,
0.040600813925266266,
-0.04181462526321411,
-0.04091596603393555,
0.03203515708446503,
0.013993220403790474,
-0.03194953128695488,
0.00893188826739788,
-0.11371342092752457,
-0.04278141260147095,
0.036711499094963074,
-0.06360315531492233,
-0.04970938712358475,
0.011640127748250961,
0.02684236876666546,
0.06074581667780876,
0.015293112024664879,
0.033892322331666946,
-0.021522466093301773,
-0.037874963134527206,
-0.10117213428020477,
0.0018107584910467267,
-0.008475768379867077,
0.045736633241176605,
-0.013208366930484772,
0.008372091688215733,
-0.11794497072696686,
0.003096070373430848,
0.07737884670495987,
-0.027159152552485466,
-0.017151879146695137,
-0.012553652748465538,
-0.013864374719560146,
0.017355963587760925,
0.05750548839569092,
0.04931959882378578,
-0.043897274881601334,
0.0018399808323010802,
0.011013716459274292,
-0.06033381447196007,
0.03566892817616463,
9.986684972100134e-32,
-0.007382337469607592,
0.008784390985965729,
-0.0634760707616806,
0.0605434887111187,
-0.0038882605731487274,
0.01935770735144615,
0.03874001279473305,
-0.030033981427550316,
0.037265416234731674,
-0.11107098311185837,
0.07609868794679642,
0.01465494092553854,
0.005105059593915939,
-0.03736408054828644,
-0.023319462314248085,
0.003732173703610897,
-0.030241699889302254,
0.10039252042770386,
-0.06575173139572144,
-0.05841381102800369,
-0.03386755287647247,
0.0479595847427845,
0.06203830987215042,
-0.055346887558698654,
0.03254833072423935,
0.07297758758068085,
-0.08128976076841354,
-0.04403289034962654,
-0.034587837755680084,
-0.04393503814935684,
0.04110103100538254,
-0.022215966135263443,
0.04262477532029152,
-0.0024895628448575735,
-0.000860617496073246,
0.014640338718891144,
-0.013433022424578667,
0.09899817407131195,
-0.014234717935323715,
0.059817519038915634,
0.024693189188838005,
-0.07304873317480087,
0.005466153845191002,
0.044818758964538574,
0.08443140238523483,
-0.05157899856567383,
-0.028106842190027237,
-0.1189684122800827,
0.08103648573160172,
-0.026621270924806595,
-0.043647658079862595,
-0.005651562009006739,
-0.037699997425079346,
-0.02865104004740715,
0.06548725813627243,
-0.01668138988316059,
-0.042412806302309036,
-0.05478433519601822,
-0.025604907423257828,
0.0031054941937327385,
0.019857775419950485,
0.10185400396585464,
-0.08275505155324936,
-0.0030981304589658976
] |
ivOverarching themes
The following three overarching themes have been fully integrated throughout the Pearson Edexcel
AS and A level Mathematics series, so they can be applied alongside your learning and practice.
1. Mathematical argument, language and proof
β’ Rigorous and consistent approach throughoutβ’ Notation boxes explain key mathematical language and symbolsβ’ Dedicated sections on mathematical proof explain key principles and strategiesβ’ Opportunities to critique arguments and justify methods
2. Mathematical problem solving
β’ Hundreds of problem-solving questions, fully integrated
into the main exercises
β’ Problem-solving boxes provide tips and strategiesβ’ Structured and unstructured questions to build confi denceβ’ Challenge boxes provide extra stretch
3. Mathematical modelling
β’ Dedicated modelling sections in relevant topics provide plenty of practice where you need it β’ Examples and exercises include qualitative questions that allow you to interpret answers in the
context of the model
β’ Dedicated chapter in Statistics & Mechanics Year 1/AS explains the principles of modelling in
mechanics Overarching themes
Each chapter starts with
a list of objectives
The Prior knowledge check
helps make sure you are ready to start the chapterThe real world applications of the maths you are about to learn are highlighted at the start of the chapter with links to relevant questions in the chapterFinding your way around the book
Access an online digital edition using the code at the front of the book.The Mathematical Problem-solving cycle
specify the problem
interpret resultscollect information
process and
represent information
|
[
0.004205680452287197,
0.09034808725118637,
-0.01995246298611164,
0.03149944543838501,
-0.011298829689621925,
0.036032337695360184,
-0.06862876564264297,
0.04851171001791954,
-0.07545731961727142,
-0.008538068272173405,
-0.08262369781732559,
-0.0017099875258281827,
-0.060057103633880615,
0.010552935302257538,
0.0661223977804184,
0.03239678964018822,
-0.043971020728349686,
0.08691967278718948,
0.00927695818245411,
-0.12467831373214722,
0.05512385442852974,
-0.01236195582896471,
-0.038049519062042236,
-0.036675211042165756,
0.020724685862660408,
-0.03124970756471157,
0.033257607370615005,
0.03574256971478462,
0.07050754129886627,
-0.1137307807803154,
-0.009154281578958035,
-0.013772038742899895,
0.1028856560587883,
-0.030254583805799484,
-0.13546395301818848,
0.0726291611790657,
0.0432521253824234,
0.1061754897236824,
-0.06740693002939224,
0.009223372675478458,
-0.048842333257198334,
-0.010716143064200878,
0.0023495671339333057,
0.01267785020172596,
0.09779836237430573,
-0.04271834343671799,
-0.01558254100382328,
-0.037113554775714874,
-0.04797140136361122,
-0.05041399970650673,
-0.02883884124457836,
-0.04325797036290169,
-0.09733156114816666,
-0.021828515455126762,
-0.029967211186885834,
-0.04290192574262619,
0.06693883240222931,
0.04229786992073059,
0.013037021271884441,
-0.020201096311211586,
0.027952490374445915,
0.013371137902140617,
-0.007645678240805864,
0.02326080948114395,
0.011155674234032631,
-0.005570245906710625,
0.034493304789066315,
0.13300184905529022,
0.0069957394152879715,
0.06931468099355698,
-0.07832212746143341,
-0.04143498092889786,
-0.012271000072360039,
0.07692287117242813,
0.09674421697854996,
0.09580215811729431,
-0.10073103755712509,
-0.09945917129516602,
0.0458049550652504,
-0.06298000365495682,
0.01991112157702446,
-0.011113250628113747,
0.0017065537394955754,
-0.00012735197378788143,
-0.030216163024306297,
-0.025370951741933823,
0.02197185531258583,
-0.027888165786862373,
-0.018579702824354172,
-0.03971412405371666,
0.12662842869758606,
-0.012934545986354351,
-0.004065573215484619,
-0.007412111386656761,
0.04188482090830803,
0.013343602418899536,
-0.11384037882089615,
-0.09984087944030762,
0.05923193693161011,
0.046835143119096756,
-0.030263489112257957,
-0.008752181194722652,
0.016442764550447464,
0.031019827350974083,
-0.07595302164554596,
0.00985515397042036,
0.055378545075654984,
-0.04807272553443909,
-0.013160181231796741,
-0.03684955835342407,
-0.03539315238595009,
-0.014263042248785496,
-0.06378880888223648,
-0.03319988399744034,
-0.02442319504916668,
-0.05146605521440506,
0.0261827502399683,
0.0074658640660345554,
0.0369887575507164,
0.03526598587632179,
-0.056971509009599686,
0.008779453113675117,
0.06284476816654205,
0.004467473365366459,
-0.017688555642962456,
0.036221880465745926,
-0.03099590726196766,
-0.03849097341299057,
-0.0031782416626811028,
-0.002075847703963518,
0.01002055499702692,
0.05036306008696556,
0.009886820800602436,
-0.08850298076868057,
0.005534152965992689,
-0.03676990047097206,
0.06393693387508392,
0.07536070793867111,
0.041233472526073456,
0.06466030329465866,
0.07137502729892731,
0.05820256099104881,
0.011619020253419876,
0.010770720429718494,
-0.07856222242116928,
-0.031219279393553734,
-0.05238998308777809,
-0.010161596350371838,
0.0330580472946167,
-0.06383726000785828,
0.0736410990357399,
-0.035674937069416046,
-0.020974377170205116,
0.06720335781574249,
0.04172559082508087,
0.04038095101714134,
0.10803592950105667,
-0.0678209736943245,
0.035947009921073914,
0.04200148209929466,
-0.04631924629211426,
0.0024348939768970013,
-0.04123691841959953,
0.05293143913149834,
-0.01571226306259632,
-0.10991688072681427,
0.054368965327739716,
-0.01747175306081772,
-0.10903792083263397,
-0.03035115636885166,
-0.03379083797335625,
0.018830692395567894,
0.005454156547784805,
0.0724504217505455,
-0.017093343660235405,
-0.05308018624782562,
0.051086701452732086,
-0.012547815218567848,
-0.05439496785402298,
-0.0014979029074311256,
0.06638053804636002,
-0.08778053522109985,
-0.019572434946894646,
-0.0190647654235363,
-0.02798384241759777,
0.01639588363468647,
0.00845523364841938,
0.015515236184000969,
-0.004947333596646786,
-0.03233102709054947,
-0.00927972886711359,
-0.020990949124097824,
-0.040788814425468445,
0.04247012734413147,
-0.07983387261629105,
-0.017578596249222755,
0.039544522762298584,
-0.03310128301382065,
-0.0430244542658329,
0.054392922669649124,
-0.05000061169266701,
-0.017364399507641792,
-0.03176254779100418,
-0.07385582476854324,
0.006486489903181791,
-0.09468801319599152,
0.03646831586956978,
-0.046439435333013535,
0.02536391094326973,
-0.005818023346364498,
0.006889604963362217,
-0.019332943484187126,
-0.002043496584519744,
-0.05739033594727516,
0.008537979796528816,
0.026193326339125633,
0.01630675606429577,
-0.043927717953920364,
-0.03368257358670235,
-0.013326828368008137,
0.03643069416284561,
-0.005779037717729807,
0.08071865886449814,
-1.568583518828099e-33,
-0.06674911081790924,
-0.016132067888975143,
-0.08257429301738739,
-0.03647841885685921,
0.027989741414785385,
0.05594708397984505,
-0.006287448573857546,
-0.08476027101278305,
0.09855050593614578,
0.00858531054109335,
-0.015267574228346348,
0.046055614948272705,
0.004658947233110666,
0.0018387497402727604,
-0.10844029486179352,
-0.14055414497852325,
-0.11917988955974579,
-0.0067297848872840405,
-0.011521607637405396,
-0.05568331480026245,
-0.03702899068593979,
0.03859243169426918,
-0.018340468406677246,
-0.09035701304674149,
-0.004407593514770269,
-0.03860316053032875,
0.016651205718517303,
-0.08358783274888992,
-0.008202447555959225,
0.02534756064414978,
0.002089044312015176,
-0.002718693343922496,
0.0002546393370721489,
0.029169779270887375,
-0.039278723299503326,
0.04230041801929474,
0.05513715371489525,
-0.051730502396821976,
0.02048301137983799,
0.011766422539949417,
-0.02781687304377556,
-0.03391742706298828,
0.0423041470348835,
-0.03292020410299301,
0.04634561017155647,
0.01933296211063862,
-0.008874843828380108,
0.016481326892971992,
-0.01572725921869278,
0.0032448931597173214,
-0.053452134132385254,
0.07652189582586288,
-0.04260024055838585,
-0.13359825313091278,
0.0767454281449318,
0.03427199646830559,
0.06391403079032898,
0.031951192766427994,
0.016552839428186417,
0.004626896232366562,
-0.02436547912657261,
0.03197232261300087,
0.07643222063779831,
0.04467960074543953,
0.09171193093061447,
-0.0053920745849609375,
-0.002898105885833502,
0.02995944768190384,
-0.03227366879582405,
0.07258488982915878,
-0.0632353350520134,
0.024919845163822174,
-0.08081845194101334,
-0.06690137833356857,
-0.017889218404889107,
0.10363306850194931,
0.03919388726353645,
0.050939805805683136,
-0.010049824602901936,
0.003686055541038513,
-0.027041342109441757,
0.027920294553041458,
-0.0072427913546562195,
0.06104814261198044,
0.019185055047273636,
0.06482487916946411,
0.06257501989603043,
-0.009101606905460358,
-0.040626153349876404,
0.017745373770594597,
0.014698974788188934,
-0.08679749071598053,
-0.03555453196167946,
-0.07509904354810715,
0.13558581471443176,
8.952494446201492e-32,
-0.02094069868326187,
-0.007979236543178558,
-0.020556844770908356,
-0.08168523013591766,
-0.03718704730272293,
0.008011783473193645,
0.05224942788481712,
-0.009004190564155579,
0.04975791648030281,
0.023891568183898926,
0.011563949286937714,
0.028597909957170486,
0.0006103620980866253,
0.048050299286842346,
-0.040449097752571106,
0.028707757592201233,
0.02137504518032074,
0.08957034349441528,
0.005097029265016317,
-0.040286678820848465,
0.06412611156702042,
0.11073444038629532,
-0.08909159898757935,
-0.057497426867485046,
-0.005008352920413017,
0.1413343995809555,
0.056168850511312485,
0.038858287036418915,
0.01758299581706524,
-0.029516704380512238,
0.04322383180260658,
-0.045123856514692307,
0.05019566789269447,
0.030274026095867157,
-0.014876899309456348,
-0.0036657939199358225,
0.05308534950017929,
0.03485879302024841,
-0.08351870626211166,
-0.044727664440870285,
-0.0704909935593605,
0.015176927670836449,
0.017752956598997116,
-0.02681518718600273,
-0.0050848969258368015,
-0.023963529616594315,
0.009794392623007298,
0.025210155174136162,
-0.04899483174085617,
-0.032264675945043564,
0.023998646065592766,
-0.0595390610396862,
-0.008930962532758713,
-0.05390743911266327,
0.07041024416685104,
0.029770389199256897,
-0.02462192252278328,
-0.003567113308236003,
-0.04831891506910324,
0.07202532142400742,
-0.004529622383415699,
0.11927475780248642,
-0.07301925122737885,
0.029828455299139023
] |
vOverarching themes
Every few chapters a Review exercise
helps you consolidate your learning with lots of exam-style questionsEach section begins
with explanation and key learning points
Step-by-step worked
examples focus on the key types of questions youβll need to tackleExercise questions are
carefully graded so they increase in diffi culty and gradually bring you up to exam standard
Problem-solving boxes provide hints, tips and strategies, and Watch out boxes highlight areas where students oft en lose marks in their examsExercises are packed with exam-style questions to ensure you are ready for the exams
A full AS level practice paper at the back of the book helps you prepare for the real thing
Exam-style questions are ο¬ agged with
Problem-solving
questions are ο¬ agged withE
PEach chapter ends with a Mixed exercise and a Summary of key pointsChallenge boxes give you a chance to tackle some more diffi cult questions
|
[
0.07345957309007645,
0.10683491826057434,
0.0449303463101387,
0.013803660869598389,
-0.0014906395226716995,
0.09178029745817184,
-0.0634760782122612,
0.05974092334508896,
-0.13185937702655792,
-0.02424071915447712,
-0.03748875856399536,
0.010927136056125164,
-0.07794016599655151,
-0.014300585724413395,
0.005704731680452824,
-0.022015748545527458,
-0.004608573392033577,
0.05603962391614914,
0.046779241412878036,
-0.06768231838941574,
0.10333804786205292,
-0.01950814761221409,
0.04500219225883484,
-0.00740129966288805,
-0.01031317375600338,
-0.018347639590501785,
0.027659529820084572,
0.0011288601672276855,
0.005596541333943605,
-0.1448521614074707,
0.006126226857304573,
0.048208821564912796,
0.07834813743829727,
0.0011417500209063292,
-0.064947709441185,
0.029906289651989937,
0.020975427702069283,
0.04823874309659004,
-0.002457048511132598,
-0.026217453181743622,
-0.09893864393234253,
0.02247474156320095,
-0.010351743549108505,
-0.01315756794065237,
0.10866262018680573,
-0.04612410068511963,
-0.04837513715028763,
-0.06137589365243912,
-0.018176592886447906,
-0.11090794950723648,
-0.045867715030908585,
-0.08369169384241104,
-0.02666744962334633,
-0.07293705642223358,
0.00030439032707363367,
0.018522758036851883,
0.04172201827168465,
0.01669463887810707,
-0.04818066954612732,
-0.015456948429346085,
-0.00453203497454524,
0.013025263324379921,
0.023359376937150955,
-0.012155946344137192,
-0.008566437289118767,
0.018281791359186172,
0.01806623488664627,
0.09669797867536545,
0.0635048896074295,
0.02343139611184597,
-0.09630360454320908,
-0.003275133902207017,
0.0346008762717247,
0.0158319640904665,
0.03551828861236572,
0.06837304681539536,
-0.14826233685016632,
-0.06963317096233368,
0.045235760509967804,
-0.022523563355207443,
-0.007164116948843002,
-0.026529327034950256,
0.0690266415476799,
-0.025396576151251793,
0.02277633175253868,
-0.03424220532178879,
-0.029217461124062538,
-0.03270140662789345,
-0.08657462149858475,
-0.028014231473207474,
0.10620976239442825,
-0.05975164845585823,
-0.03059113398194313,
-0.035717107355594635,
0.02720903605222702,
0.0675329938530922,
-0.0371236726641655,
-0.05009755492210388,
0.06991884857416153,
0.007639887742698193,
0.05404314771294594,
0.04226486012339592,
-0.0010661996202543378,
0.08298447728157043,
-0.1012808233499527,
0.03541138023138046,
0.04164540022611618,
-0.04925728961825371,
0.03856687247753143,
-0.07080566883087158,
-0.014969853684306145,
-0.04691245034337044,
-0.05914326757192612,
-0.08441078662872314,
0.0198648851364851,
0.020661409944295883,
0.019327668473124504,
0.08753186464309692,
-0.06270932406187057,
0.0694349855184555,
0.03392346575856209,
-0.048709042370319366,
0.015361594967544079,
-0.08017772436141968,
0.07931876182556152,
0.03238210082054138,
0.021443655714392662,
-0.01095323171466589,
0.05780298262834549,
-0.014908545650541782,
-0.016036489978432655,
0.13203100860118866,
-0.010657105594873428,
-0.02373884990811348,
0.021572206169366837,
-0.011130645871162415,
0.0033561247400939465,
0.035951342433691025,
0.03551172837615013,
0.042734336107969284,
-0.010694335214793682,
0.06678382307291031,
0.02973228693008423,
0.01551899965852499,
-0.0904705822467804,
0.03077664226293564,
-0.033293262124061584,
0.002943295519798994,
0.0014332961291074753,
-0.08308884501457214,
-0.0027909819036722183,
-0.06593607366085052,
0.064442478120327,
0.03438100218772888,
0.03442968800663948,
0.016380811110138893,
-0.016094258055090904,
-0.03416424244642258,
0.07359981536865234,
-0.01118288654834032,
-0.06716029345989227,
0.05878995731472969,
0.026873502880334854,
0.04284873232245445,
0.044698573648929596,
0.01562640070915222,
0.03503933921456337,
-0.060080479830503464,
-0.053849026560783386,
-0.05521615967154503,
0.049572236835956573,
-0.029950421303510666,
0.015742305666208267,
0.05161529406905174,
0.039879150688648224,
-0.01100962795317173,
0.07243221998214722,
0.007248356938362122,
-0.04699157550930977,
-0.04764896631240845,
0.08693491667509079,
-0.08108691871166229,
-0.03893159702420235,
-0.01514100469648838,
0.08217646181583405,
-0.020523052662611008,
-0.028264131397008896,
-0.06506462395191193,
0.0595732145011425,
0.028071753680706024,
-0.033926937729120255,
-0.043821025639772415,
-0.05397013574838638,
-0.010487827472388744,
-0.10171832144260406,
-0.03217942640185356,
0.02285328507423401,
-0.0637742131948471,
-0.10984089970588684,
-0.0038465766701847315,
-0.02705945447087288,
-0.05298362672328949,
-0.0015895836986601353,
0.01957550086081028,
0.012309153564274311,
-0.06230495497584343,
0.05823073908686638,
-0.03441495820879936,
0.14851199090480804,
0.007305174134671688,
-0.026979468762874603,
-0.027445826679468155,
0.018697552382946014,
-0.019406691193580627,
0.04261111468076706,
-0.00673397071659565,
0.02873600274324417,
-0.10354912281036377,
-0.04003334045410156,
-0.014041490852832794,
0.08203592896461487,
-0.007466943934559822,
0.07063984125852585,
3.538219628875854e-33,
-0.04353351891040802,
0.012885535135865211,
-0.10873112082481384,
0.03902491182088852,
0.03656422346830368,
0.02133173868060112,
0.0070616998709738255,
-0.01300059910863638,
0.0201751459389925,
0.027432026341557503,
-0.028215890750288963,
-0.05162721872329712,
0.015615631826221943,
0.02013835869729519,
-0.05264570936560631,
-0.10381168127059937,
-0.04673938825726509,
0.03763960301876068,
-0.0007880758494138718,
-0.09444309771060944,
0.005030914209783077,
0.11268558353185654,
0.048550721257925034,
-0.05791434273123741,
-0.015035503543913364,
-0.020430931821465492,
0.011814991012215614,
-0.014744697138667107,
0.041198838502168655,
0.03462965786457062,
0.03240610286593437,
-0.028389131650328636,
0.0103730708360672,
0.008520352654159069,
-0.07693043351173401,
0.03180608153343201,
0.04797503352165222,
-0.039796844124794006,
0.04345841333270073,
0.02392546646296978,
-0.0019024143693968654,
-0.004001993220299482,
0.006004284135997295,
-0.053681064397096634,
-0.014164737425744534,
-0.00655654352158308,
0.012019569054245949,
0.018755005672574043,
-0.07504617422819138,
0.019621271640062332,
0.03025730885565281,
0.06084571033716202,
-0.07741548120975494,
-0.04207657277584076,
0.06054973974823952,
0.027493800967931747,
0.03443504497408867,
-0.0478455051779747,
-0.04781010001897812,
-0.03082510642707348,
-0.022764870896935463,
0.00906927790492773,
0.016886206343770027,
0.033602554351091385,
0.07043961435556412,
-0.04866436868906021,
0.025425542145967484,
-0.0019932982977479696,
-0.053472742438316345,
-0.01208541076630354,
-0.13747745752334595,
-0.007178331725299358,
-0.006086068227887154,
-0.044545553624629974,
0.018498504534363747,
0.02990017458796501,
0.08047246932983398,
-0.014912786893546581,
-0.05928363651037216,
-0.023759989067912102,
0.006950851995497942,
-0.031997308135032654,
-0.0007817773730494082,
0.018548183143138885,
0.019435996189713478,
0.08867044001817703,
0.004657408222556114,
0.0620332695543766,
0.0274299755692482,
-0.01748836599290371,
-0.010800371877849102,
-0.11794769018888474,
0.059874460101127625,
-0.0765281617641449,
0.03236459195613861,
6.466459663809874e-32,
-0.050117526203393936,
-0.04070434719324112,
-0.007876468822360039,
0.03912921994924545,
-0.06747936457395554,
-0.021372614428400993,
0.020798727869987488,
0.00805687066167593,
0.044561464339494705,
0.08369776606559753,
0.03212849423289299,
0.010034442879259586,
-0.008985173888504505,
0.07614462822675705,
0.04711545631289482,
0.09863874316215515,
-0.0214335098862648,
0.1339646577835083,
-0.02767804078757763,
-0.09121192246675491,
0.06535284966230392,
-0.0031923737842589617,
-0.03258218616247177,
-0.06203782930970192,
-0.010367105714976788,
0.02507566101849079,
0.03961644694209099,
-0.026129089295864105,
-0.04683934152126312,
0.012047759257256985,
0.028516575694084167,
0.0015647923573851585,
0.01877458021044731,
0.007090471219271421,
-0.02744627557694912,
0.012650813907384872,
0.05014423280954361,
-0.013853971846401691,
0.02303921990096569,
0.06705562770366669,
-0.08741258084774017,
-0.010118725709617138,
0.10745295882225037,
-0.024527007713913918,
-0.024526074528694153,
-0.06704588979482651,
-0.08068666607141495,
0.02040702849626541,
-0.07594212144613266,
-0.00019572391465771943,
0.052417781203985214,
-0.03321891278028488,
0.031050706282258034,
-0.08594167977571487,
0.040801625698804855,
0.11267147213220596,
-0.0036316330078989267,
0.017956605181097984,
-0.016755152493715286,
0.06973759829998016,
0.03683329001069069,
0.09217298030853271,
-0.06547998636960983,
0.01849856786429882
] |
viExtra online content
Whenever you see an Online box, it means that there is extra online content available to support you.
SolutionBank
SolutionBank provides a full worked solution for
every question in the book.
Download all the solutions
as a PDF or quickly fi nd the solution you need online Extra online content
Full worked solutions are
available in SolutionBank.Online
|
[
-0.01650870405137539,
-0.04769827798008919,
0.02113061212003231,
-0.02006075717508793,
0.10368504375219345,
0.0038298163563013077,
-0.10987254232168198,
0.09201961755752563,
-0.056575797498226166,
0.012711485847830772,
0.009425300173461437,
0.011357325129210949,
0.0173902940005064,
0.03134553134441376,
0.033186428248882294,
-0.08311718702316284,
0.05425608158111572,
0.03138047829270363,
-0.007123925723135471,
-0.021298957988619804,
0.06733209639787674,
-0.038808610290288925,
-0.01929529197514057,
-0.009148312732577324,
0.035045355558395386,
-0.03724953532218933,
-0.03468109294772148,
0.05179695412516594,
0.06391959637403488,
-0.012526463717222214,
0.03272711858153343,
0.023989630863070488,
0.01988021843135357,
-0.03516944497823715,
-0.117819644510746,
-0.02271880954504013,
0.004448718391358852,
0.045669157058000565,
-0.005148476455360651,
-0.04815362021327019,
0.09128625690937042,
0.022058673202991486,
-0.04830991476774216,
0.015627959743142128,
0.03091958537697792,
-0.0978168249130249,
-0.053723279386758804,
0.05708494782447815,
0.03297778218984604,
-0.06665016710758209,
0.030693866312503815,
-0.006061200052499771,
-0.004976476542651653,
0.0016370355151593685,
-0.06443817913532257,
-0.01785651221871376,
0.06409384310245514,
0.07637351751327515,
-0.046568937599658966,
-0.003586933948099613,
0.08608509600162506,
-0.012917887419462204,
-0.018571438267827034,
0.07864149659872055,
-0.005852816626429558,
0.02558993361890316,
-0.0879756286740303,
0.12825658917427063,
0.010885071009397507,
-0.0568491593003273,
-0.07915959507226944,
-0.03532302379608154,
0.028953539207577705,
0.07312764972448349,
0.07867135107517242,
0.007602772209793329,
-0.008458878844976425,
-0.0612504705786705,
-0.043773047626018524,
0.04684856906533241,
0.03479385003447533,
0.011174836196005344,
0.06396590918302536,
-0.08367925137281418,
-0.07476244121789932,
0.07353627681732178,
0.034591469913721085,
0.026397785171866417,
0.0758247897028923,
0.0633251741528511,
-0.002578833606094122,
-0.014231426641345024,
0.02563493698835373,
-0.032483961433172226,
0.06449262797832489,
-0.003155349986627698,
-0.05557809770107269,
-0.022136671468615532,
0.08089771866798401,
0.03128020837903023,
-0.01824653334915638,
0.08028831332921982,
0.10035734623670578,
-0.030851582065224648,
-0.039245206862688065,
0.021276643499732018,
0.08166073262691498,
0.11357284337282181,
0.05481657758355141,
-0.04456763714551926,
-0.09746234118938446,
-0.0431034117937088,
-0.05465728044509888,
-0.11577657610177994,
0.04000382125377655,
0.02119048871099949,
-0.023478053510189056,
-0.03735100477933884,
0.01654203049838543,
0.04527153819799423,
-0.08319460600614548,
0.06611757725477219,
0.06578958034515381,
-0.05241885408759117,
0.1388952136039734,
-0.06583380699157715,
0.05931256711483002,
-0.003835319308564067,
-0.023022696375846863,
0.008293681778013706,
0.053580187261104584,
-0.019664781168103218,
-0.07884451001882553,
-0.015922565013170242,
-0.007893512025475502,
-0.0012691023293882608,
0.029085304588079453,
0.04425881803035736,
0.03169885650277138,
0.05374220758676529,
-0.021201925352215767,
0.018296275287866592,
-0.032366905361413956,
-0.0326242633163929,
-0.047484274953603745,
0.014322073198854923,
0.0792648047208786,
0.08823756873607635,
-0.0015204233350232244,
-0.07604116946458817,
0.01842111349105835,
-0.06743334978818893,
-0.006401250138878822,
0.06030246615409851,
0.012771339155733585,
0.0015694507164880633,
0.12026689946651459,
-0.06265134364366531,
-0.07851453870534897,
0.01912512443959713,
-0.02875729836523533,
0.03416864946484566,
-0.0265438724309206,
0.015115916728973389,
-0.09549462050199509,
-0.013890470378100872,
0.04630003869533539,
0.00486145680770278,
-0.06950481235980988,
-0.026850830763578415,
-0.018664898350834846,
0.01711737923324108,
-0.01763637736439705,
0.029451239854097366,
0.07144615799188614,
-0.06066169962286949,
-0.01634836383163929,
0.022186895832419395,
-0.02068273350596428,
-0.013544308952987194,
-0.07124149799346924,
0.009518053382635117,
-0.09256525337696075,
0.07769414782524109,
-0.08942107856273651,
-0.0024202691856771708,
0.053287360817193985,
-0.04101823270320892,
0.032118599861860275,
-0.051968544721603394,
-0.03929651528596878,
-0.06458652019500732,
0.0588553287088871,
0.04557519778609276,
0.021036827936768532,
-0.08097352087497711,
-0.09335821121931076,
-0.040575459599494934,
-0.07394513487815857,
0.013713848777115345,
-0.0038668951019644737,
0.09740012884140015,
-0.011685090139508247,
-0.0290738083422184,
-0.006891907658427954,
0.033916521817445755,
-0.02582930028438568,
-0.055059533566236496,
-0.04144500195980072,
0.008381437510251999,
-0.03126097097992897,
0.042621370404958725,
0.03797873854637146,
0.12135713547468185,
0.050738878548145294,
0.0003628442354965955,
0.008344327099621296,
-0.059962958097457886,
-0.0781608372926712,
0.026648763567209244,
-0.07109104841947556,
-0.1218716949224472,
0.05548635870218277,
2.485369057696419e-34,
-0.07670453190803528,
-0.09266465157270432,
-0.07048942893743515,
0.021498210728168488,
0.047969378530979156,
-0.010420108214020729,
0.0162999015301466,
-0.031116340309381485,
0.11040546000003815,
0.0512443445622921,
-0.0032605333253741264,
-0.01395601686090231,
0.03938740864396095,
0.019856275990605354,
-0.08265905827283859,
0.05026208981871605,
-0.014854956418275833,
-0.03361948952078819,
0.08442211896181107,
-0.022997956722974777,
0.022432364523410797,
0.024958379566669464,
-0.06729419529438019,
-0.0742432028055191,
0.10437721014022827,
-0.0006000449066050351,
0.036276962608098984,
-0.06772804260253906,
-0.010296915657818317,
0.09276256710290909,
-0.00031642502290196717,
-0.07949621230363846,
0.00038431212306022644,
-0.003442224347963929,
-0.0163030494004488,
-0.02770889922976494,
-0.044483255594968796,
-0.03757674992084503,
0.04892997071146965,
-0.0393318273127079,
0.0315844863653183,
-0.00739767262712121,
-0.09545860439538956,
-0.048705846071243286,
-0.06498565524816513,
-0.028172146528959274,
-0.06445267796516418,
-0.036132052540779114,
-0.02049919217824936,
-0.013107885606586933,
0.0504070445895195,
0.008749577216804028,
0.02196313627064228,
-0.04605095088481903,
0.028157483786344528,
0.09148164093494415,
0.025716980919241905,
0.0020320387557148933,
-0.00029660656582564116,
0.0013118492206558585,
-0.05357646569609642,
-0.011511305347084999,
0.010599246248602867,
0.030281195417046547,
0.07014858722686768,
-0.00982610508799553,
0.002462402218952775,
0.06319937855005264,
-0.04121449962258339,
-0.006591752637177706,
-0.0660015419125557,
-0.0701516792178154,
0.041419826447963715,
0.008822294883430004,
-0.03552888333797455,
0.041371919214725494,
-0.050030067563056946,
0.009124945849180222,
-0.021286282688379288,
-0.020526405423879623,
-0.02059238962829113,
-0.007496607955545187,
0.027545761317014694,
-0.008558295667171478,
0.08187826722860336,
-0.0024929943028837442,
-0.04934430494904518,
0.04846890643239021,
0.0059689623303711414,
0.04320041835308075,
-0.00827585905790329,
-0.045207053422927856,
0.005355048459023237,
0.07355380803346634,
0.0074870530515909195,
8.227700498658206e-32,
0.028752410784363747,
-0.06832125037908554,
-0.07488065958023071,
0.00707820663228631,
0.03578634187579155,
-0.004477847367525101,
0.02902098372578621,
0.010714403353631496,
0.05964895710349083,
-0.057682331651449203,
-0.08374326676130295,
-0.051067691296339035,
-0.022045230492949486,
0.01223402377218008,
0.01247680839151144,
0.03431471437215805,
-0.09787071496248245,
0.07141008973121643,
0.013153689913451672,
-0.0814870223402977,
-0.03282984346151352,
0.04590342193841934,
-0.00719818938523531,
-0.022578371688723564,
-0.014595930464565754,
0.011447137221693993,
0.007134781684726477,
0.02658940851688385,
-0.008601784706115723,
-0.031903453171253204,
0.09371238946914673,
-0.013326052576303482,
0.053442392498254776,
-0.0699465274810791,
0.04962189868092537,
-0.03708070516586304,
-0.005254870280623436,
-0.012721441686153412,
-0.10151007771492004,
0.051346659660339355,
-0.048131611198186874,
-0.039991073310375214,
0.0068114567548036575,
-0.08052364736795425,
0.042655546218156815,
0.01968349702656269,
0.028165588155388832,
-0.10170597583055496,
0.07449804991483688,
-0.0877876952290535,
-0.03505197539925575,
-0.015849009156227112,
0.06977474689483643,
-0.054568592458963394,
0.054878801107406616,
-0.0034486178774386644,
-0.03884264826774597,
-0.039182692766189575,
0.086812324821949,
-0.013466497883200645,
-0.005849751643836498,
0.05060799419879913,
0.03256648778915405,
-0.017550591379404068
] |
viiExtra online content
Access all the extra online content for FREE at:
www.pearsonschools.co.uk/p1maths
You can also access the extra online content by scanning this QR Code:
GeoGebra interactives
Explore topics in more detail,
visualise problems and consolidate your understanding with GeoGebra-powered interactives.
Interact with the maths
you are learning using GeoGebra's easy-to-use tools
Explore the gradient of the
chord AP using GeoGebra.Online
Casio calculator support
Our helpful tutorials will guide
you through how to use your calculator in the exams. They cover both Casio's scientific and colour graphic calculators.
See exactly which
buttons to press and what should appear on your calculator's screen
Work out each coefficient
qui
ckly using the nCr and power
functions on your calculator.Online
|
[
-0.028821848332881927,
-0.02600421942770481,
0.0006132807466201484,
-0.08141008764505386,
-0.06986618041992188,
0.040282171219587326,
-0.09036553651094437,
0.09533381462097168,
-0.05214819312095642,
0.01122585404664278,
-0.04918069392442703,
-0.007434755563735962,
0.0008915112121030688,
0.03233213722705841,
0.023627404123544693,
-0.04824095964431763,
0.04203210771083832,
0.05769152194261551,
-0.01642664149403572,
-0.015479980036616325,
0.10359548777341843,
-0.05319267883896828,
-0.0680060088634491,
-0.03219493851065636,
0.033392734825611115,
0.005186708178371191,
-0.008816555142402649,
0.031565796583890915,
0.021589338779449463,
-0.02188488468527794,
0.02500866912305355,
0.042295150458812714,
0.132557213306427,
-0.07321349531412125,
-0.13027328252792358,
-0.056416891515254974,
0.042642753571271896,
0.006875287741422653,
-0.03908069431781769,
-0.010790175758302212,
-0.028876623138785362,
0.07986066490411758,
-0.06135118380188942,
0.04946639761328697,
0.055756717920303345,
-0.049824535846710205,
-0.02470063976943493,
-0.07926887273788452,
0.045616600662469864,
-0.043424468487501144,
0.003499126061797142,
-0.014451316557824612,
-0.07475682348012924,
0.003004473401233554,
-0.00103392219170928,
0.003624304663389921,
-0.00813035387545824,
0.09069819003343582,
0.06178390979766846,
-0.01921236142516136,
0.059167370200157166,
-0.030025992542505264,
0.01997288502752781,
0.050124429166316986,
-0.04816626012325287,
-0.01775304600596428,
0.06179923936724663,
0.05167698860168457,
0.017973406240344048,
-0.054740022867918015,
-0.08415746688842773,
-0.05408059433102608,
0.019876452162861824,
0.04662435129284859,
0.056588128209114075,
0.013292611576616764,
-0.07376807183027267,
-0.04483971744775772,
-0.02424795553088188,
-0.038601890206336975,
0.020408157259225845,
-0.026168813928961754,
-0.015806013718247414,
-0.05466945096850395,
0.04571528360247612,
0.011408240534365177,
0.08157280087471008,
0.02486688643693924,
0.07519073039293289,
0.03773340955376625,
0.12424955517053604,
-0.06417692452669144,
-0.10862545669078827,
-0.029782487079501152,
-0.08356769382953644,
-0.03175077587366104,
-0.025937138125300407,
-0.03488621860742569,
0.025989200919866562,
0.07066728174686432,
0.056402646005153656,
0.007275090552866459,
0.06086680665612221,
0.026459908112883568,
-0.03921648859977722,
-0.007074641063809395,
0.11158481240272522,
0.007003785111010075,
0.06817235052585602,
-0.055094946175813675,
-0.058102525770664215,
-0.08974775671958923,
-0.020685546100139618,
-0.0961548238992691,
0.0028418987058103085,
-0.0182709489017725,
0.05101203918457031,
0.008967875503003597,
0.14021874964237213,
0.020096248015761375,
-0.06260963529348373,
0.017688505351543427,
0.07225706428289413,
0.07285360246896744,
0.07082581520080566,
-0.06466224789619446,
-0.05600301921367645,
0.03714701160788536,
-0.017558790743350983,
0.004358618054538965,
0.08905605971813202,
0.057388730347156525,
-0.02485845983028412,
-0.015004185028374195,
-0.03874282166361809,
-0.034566935151815414,
0.02986702136695385,
0.004983861930668354,
-0.03228560835123062,
0.06408514827489853,
0.05920633301138878,
0.03482968732714653,
-0.05742350593209267,
-0.06965462863445282,
-0.07990176230669022,
-0.03149566426873207,
0.04241693392395973,
0.02459118887782097,
-0.023728201165795326,
-0.09481634199619293,
-0.048922132700681686,
0.016070622950792313,
-0.042615026235580444,
0.12722517549991608,
0.07703414559364319,
0.03510172292590141,
0.09955863654613495,
-0.020150868222117424,
-0.01971987821161747,
-0.07556405663490295,
-0.09806299954652786,
0.03520992398262024,
0.012642290443181992,
0.09208648651838303,
0.019869573414325714,
0.004572033882141113,
0.08716993033885956,
-0.026407720521092415,
-0.017126770690083504,
0.02565520443022251,
-0.0364900603890419,
0.043533436954021454,
0.006526276934891939,
-0.07469148188829422,
0.06474651396274567,
-0.011794030666351318,
0.05911318212747574,
0.02688642218708992,
-0.08176594972610474,
-0.012163600884377956,
-0.03401491791009903,
-0.0006408898043446243,
-0.044176921248435974,
0.000872712058480829,
-0.0029591417405754328,
0.048968251794576645,
0.005550116766244173,
0.00485187117010355,
0.03228134289383888,
0.033957600593566895,
0.03633375093340874,
-0.04918947443366051,
-0.039124902337789536,
0.011701417155563831,
-0.08410526812076569,
-0.03505909442901611,
-0.0872221291065216,
-0.0004850230470765382,
-0.07511613517999649,
0.04883447289466858,
0.06269152462482452,
-0.008856910280883312,
0.0008157877600751817,
-0.01336607988923788,
-0.06204045191407204,
-0.0646301880478859,
0.06562594324350357,
-0.10657144337892532,
-0.027487900108098984,
0.012151820585131645,
0.029744431376457214,
-0.013794280588626862,
0.07719121873378754,
-0.04638216644525528,
0.03976520523428917,
-0.03702961653470993,
-0.023731540888547897,
-0.03362749144434929,
-0.11494142562150955,
-0.006885703653097153,
-0.0788949728012085,
-0.02240046113729477,
0.1009572297334671,
4.9980113292063425e-33,
-0.07907414436340332,
0.01152954064309597,
-0.0784764364361763,
0.022421913221478462,
0.009053629823029041,
-0.006928909104317427,
0.030916329473257065,
0.008884808048605919,
0.05774061754345894,
0.01243265625089407,
0.040119051933288574,
0.012150612659752369,
0.06204313412308693,
-0.017916109412908554,
0.0013356618583202362,
-0.005973604507744312,
-0.0746987909078598,
0.013659022748470306,
-0.025323593989014626,
-0.000264364032773301,
-0.05790644511580467,
0.050608500838279724,
-0.04173988476395607,
0.005342308897525072,
0.03695197030901909,
-0.011955761350691319,
0.036957625299692154,
-0.03545771911740303,
0.004172400571405888,
0.04538604989647865,
-0.033813100308179855,
-0.07988574355840683,
-0.0007319838041439652,
0.02037610299885273,
-0.122403584420681,
-0.00649374770000577,
0.021217070519924164,
-0.029163306578993797,
0.020806824788451195,
-0.11259002238512039,
0.01378518808633089,
0.0005718189058825374,
0.004398645833134651,
-0.01235212292522192,
-0.021413221955299377,
0.06855525821447372,
0.018098656088113785,
0.11424343287944794,
0.02396306023001671,
0.04130103066563606,
-0.003187558613717556,
0.006717142648994923,
0.03584401682019234,
-0.04864591732621193,
0.07443159818649292,
0.027782384306192398,
-0.049090951681137085,
-0.0695369765162468,
0.047189049422740936,
0.02582111768424511,
-0.039360810071229935,
-0.12432277947664261,
-0.0545303151011467,
0.029073502868413925,
0.06009601056575775,
0.045918792486190796,
0.03984038159251213,
0.08508019894361496,
-0.048215072602033615,
0.011639928445219994,
-0.00008565667667426169,
0.030019870027899742,
0.03688337653875351,
-0.10337082296609879,
-0.044884972274303436,
-0.003677964676171541,
0.04174404218792915,
0.05695426091551781,
-0.05563544109463692,
-0.08982869982719421,
0.0532928965985775,
-0.009394777938723564,
0.011129011400043964,
-0.10027473419904709,
0.019691865891218185,
0.05676552280783653,
-0.03029617667198181,
0.06284009665250778,
-0.058238811790943146,
-0.05810239166021347,
-0.013274149969220161,
0.03711669147014618,
0.012164812535047531,
-0.0708584114909172,
-0.012372276745736599,
8.32888176265241e-32,
-0.020728416740894318,
0.013740254566073418,
-0.07753296196460724,
-0.06848971545696259,
-0.03940850868821144,
-0.013157269917428493,
0.004874527920037508,
-0.0003736456565093249,
0.0003017241251654923,
-0.03651397302746773,
-0.01868748664855957,
-0.01769113354384899,
-0.04772913083434105,
0.05811547115445137,
-0.026013869792222977,
0.04705241695046425,
0.0017992962384596467,
0.07127221673727036,
-0.010815391317009926,
0.009961152449250221,
-0.016023686155676842,
0.06062974035739899,
0.07252911478281021,
-0.00935455784201622,
0.03060649149119854,
0.04457879066467285,
-0.00866137258708477,
0.06264564394950867,
-0.020283395424485207,
-0.06640413403511047,
0.056133367121219635,
0.012172695249319077,
0.045922670513391495,
-0.06376034766435623,
0.000899390026461333,
0.002080742735415697,
0.013941848650574684,
0.017582371830940247,
-0.03617453575134277,
0.11089231073856354,
0.05974670127034187,
-0.11868275701999664,
0.027003400027751923,
0.02290157601237297,
0.052243251353502274,
0.013370807282626629,
-0.005080170929431915,
-0.03762052208185196,
0.029308829456567764,
-0.04623040929436684,
0.020684774965047836,
-0.02826690301299095,
-0.05261069908738136,
-0.056033432483673096,
0.07982425391674042,
-0.01801919937133789,
-0.021864963695406914,
-0.030717195942997932,
0.03841732069849968,
0.07009690999984741,
-0.041870519518852234,
0.07093711942434311,
0.005083572119474411,
-0.04079055413603783
] |
viiiPublished by Pearson Education Limited, 80 Strand, London WC2R 0RL.
www.pearsonschoolsandfecolleges.co.uk Copies of official specifications for all Pearson qualifications may be found on the website:
qualifications.pearson.com
Text Β© Pearson Education Limited 2017
Edited by Tech-Set Ltd, GatesheadTypeset by Tech-Set Ltd, GatesheadOriginal illustrations Β© Pearson Education Limited 2017 Cover illustration Marcus@kja-artists
The rights of Greg Attwood, Jack Barraclough, Ian Bettison, Alistair Macpherson, Bronwen
Moran, Su Nicholson, Diane Oliver, Joe Petran, Keith Pledger, Harry Smith, Geoff Staley, RobertΒ Ward
-Penny, Dave Wilkins to be identified as authors of this work have been asserted
by them in accordance with the Copyright, Designs and Patents Act 1988.
First published 201720 19 18 17
10 9 8 7 6 5 4 3 2 1
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
ISBN 978 1 292 20826 8 (Print)
Copyright notice
All rights reserved. No part of this publication may be reproduced in any form or by any means (including photocopying or storing it in any medium by electronic means and whether or not transiently or incidentally to some other use of this publication) without the written permission of the copyright owner, except in accordance with the provisions of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency, Barnards Inn 86 Fetter Lane, London EC4A 1EN (www.cla.co.uk). Applications for the copyright ownerβs written permission should be addressed to the publisher.
Printed in Slovakia by NeografiaPicture Credits
The publisher would like to thank the following for their kind permission to reproduce their photographs:
(Key: b-bottom; c-centre; l-left; r-right; t-top)123RF.com: David Acosta Allely 287, 338cr; Alamy Images: Utah Images 113, 226l, Xinhua 38,
85cr, ZUMA Press, Inc. 311, 338r; Fotolia.com: Kajano 137, 226cl, sborisov 173, 226r, Thaut Images 202, 226tr; Getty Images: Graiki 255, 338cl, Henglein and Steets 18, 85c, Jeff Schultz 230, 338l, mviamonte 1, 85l, Steve Dunwell 158, 226cr; Science Photo Library Ltd: CMS EXPERIMENT, CERN 59, 85; Shutterstock.com: vladimir salman 89, 226tl
All other images Β© Pearson EducationISBN 978 1 292 20759 9 (PDF)
|
[
0.06531123071908951,
0.019349830225110054,
0.07931661605834961,
0.022905973717570305,
0.04012357443571091,
-0.06148500367999077,
-0.12610028684139252,
-0.015299461781978607,
-0.12173676490783691,
0.028318144381046295,
0.00506283063441515,
0.004291235934942961,
-0.003583521582186222,
0.05338042601943016,
-0.028471529483795166,
-0.007668706588447094,
-0.0538175106048584,
-0.003444342641159892,
0.03692474961280823,
-0.04752156510949135,
-0.04993271455168724,
-0.02568584494292736,
0.029086031019687653,
-0.02806318737566471,
0.044606879353523254,
0.02923513948917389,
-0.06991222500801086,
-0.026970872655510902,
0.0023427470587193966,
-0.025148063898086548,
0.010699578560888767,
-0.06184433400630951,
0.029278622940182686,
0.0025670966133475304,
-0.07997646182775497,
0.04231259599328041,
0.056718550622463226,
-0.0033325108233839273,
-0.016992514953017235,
-0.09925305843353271,
-0.04811101406812668,
-0.07129158079624176,
-0.0058168875984847546,
0.040216878056526184,
0.06767310202121735,
0.04880598932504654,
-0.06587357074022293,
-0.053882621228694916,
-0.09871882200241089,
-0.07102333009243011,
-0.03953763097524643,
-0.03796876594424248,
-0.02321815676987171,
-0.05254480987787247,
-0.060810428112745285,
-0.06598088145256042,
0.013747244141995907,
0.1026768684387207,
-0.01744362898170948,
-0.02537314034998417,
-0.045113734900951385,
0.04675610363483429,
-0.024246005341410637,
0.0631859079003334,
0.05901581794023514,
0.0236112792044878,
-0.006658619269728661,
0.11169886589050293,
0.026720119640231133,
-0.04092616215348244,
-0.08775873482227325,
-0.03203370049595833,
0.004636034835129976,
0.09586183726787567,
0.05476591736078262,
0.039210569113492966,
-0.024594999849796295,
-0.020457051694393158,
0.09160242974758148,
-0.05756096914410591,
0.023355912417173386,
0.06929326802492142,
-0.02339620515704155,
0.025517743080854416,
-0.008744527585804462,
-0.03919539600610733,
-0.004941191058605909,
0.046650230884552,
-0.04608626291155815,
0.040956541895866394,
0.08055386692285538,
-0.05536365509033203,
-0.07804839313030243,
0.05749700963497162,
0.0265941359102726,
-0.028957145288586617,
-0.04839901626110077,
-0.05440577119588852,
0.0010199954267591238,
-0.0021666809916496277,
-0.0906745195388794,
-0.04750203341245651,
0.03803800791501999,
0.00988782662898302,
-0.0640556663274765,
0.05000497028231621,
-0.021054502576589584,
0.007008351851254702,
0.042194873094558716,
-0.05161922797560692,
0.011643058620393276,
-0.06080888584256172,
-0.02205014042556286,
-0.06943369656801224,
-0.007216159719973803,
-0.006489839870482683,
-0.04694153741002083,
0.03429342433810234,
0.05254048481583595,
0.06411368399858475,
-0.01151201967149973,
0.04641377180814743,
0.00997940544039011,
-0.12404774874448776,
-0.02835187502205372,
0.004008775111287832,
-0.0028754507657140493,
0.0351019948720932,
-0.02774314396083355,
0.07891446352005005,
-0.0365561917424202,
0.056628622114658356,
0.02760973758995533,
0.02576359547674656,
0.0008918257663026452,
0.02380509115755558,
-0.00034377307747490704,
0.07490499317646027,
0.025133302435278893,
0.08492300659418106,
0.045220762491226196,
0.04430960491299629,
-0.028302039951086044,
-0.04894261434674263,
-0.04614365100860596,
0.010477445088326931,
-0.06665270030498505,
0.09539845585823059,
-0.03613210842013359,
-0.005121320486068726,
0.02865970879793167,
0.04320955276489258,
-0.01055740937590599,
0.06540665775537491,
0.0316014401614666,
-0.020223001018166542,
0.11375213414430618,
0.04437669739127159,
0.1205056682229042,
-0.07529877871274948,
-0.06933615356683731,
-0.05066553130745888,
0.05462607368826866,
-0.0705394372344017,
0.015146797522902489,
-0.0888267308473587,
0.10437914729118347,
-0.006526549346745014,
-0.06821359694004059,
0.03867234289646149,
-0.0522967204451561,
-0.022843772545456886,
0.03143123909831047,
0.040730610489845276,
0.0359983891248703,
-0.04934217035770416,
-0.015710312873125076,
0.07671912759542465,
-0.02873171865940094,
-0.02411252073943615,
-0.013396011665463448,
-0.047546546906232834,
0.011051958426833153,
-0.0557546466588974,
0.12826921045780182,
0.030021095648407936,
0.0022432238329201937,
-0.11504445970058441,
0.07949157059192657,
-0.005024388898164034,
-0.03445998579263687,
-0.0798191949725151,
-0.015895839780569077,
-0.04027966782450676,
0.03572118654847145,
-0.02019215188920498,
0.010173197835683823,
-0.05552047863602638,
-0.054829586297273636,
0.015635589137673378,
-0.036117225885391235,
-0.01933903805911541,
0.007066778372973204,
0.07814882695674896,
-0.025599341839551926,
-0.04482417553663254,
0.07434192299842834,
-0.009042767807841301,
0.03943753242492676,
0.05639356002211571,
-0.00874099601060152,
-0.12178272008895874,
0.011615010909736156,
-0.0753156915307045,
0.013686643913388252,
-0.039825309067964554,
0.02979482151567936,
-0.018950317054986954,
0.06901492178440094,
-0.027557166293263435,
-0.10297849029302597,
0.03100535087287426,
0.06865135580301285,
-2.826706673647547e-33,
-0.014809138141572475,
0.011289895512163639,
-0.03116208128631115,
-0.010062528774142265,
0.05461161211133003,
-0.03367475047707558,
0.016853852197527885,
-0.0646466389298439,
0.09525983780622482,
-0.03596242144703865,
0.025495164096355438,
0.003350568236783147,
-0.001066309050656855,
0.03575003519654274,
-0.009762363508343697,
-0.035155076533555984,
-0.14491058886051178,
-0.03057880513370037,
-0.10952383279800415,
-0.07262203097343445,
0.014934196136891842,
-0.06517311185598373,
-0.03868348151445389,
-0.050619110465049744,
0.01069382019340992,
-0.046776607632637024,
0.0011639404110610485,
-0.03778480365872383,
-0.06745705753564835,
0.022033782675862312,
0.01565011404454708,
0.009190469048917294,
-0.009626258164644241,
-0.039738576859235764,
-0.09045856446027756,
-0.07627270370721817,
-0.008473255671560764,
-0.06698321551084518,
-0.011170495301485062,
0.03853815793991089,
0.039039246737957,
0.0028277398087084293,
-0.08094381541013718,
-0.030059058219194412,
-0.045606579631567,
0.004742391873151064,
0.0168928150087595,
-0.00706096226349473,
-0.015981514006853104,
-0.023343540728092194,
0.006789051461964846,
0.040795981884002686,
-0.03394485265016556,
-0.09224284440279007,
0.13235652446746826,
-0.019666263833642006,
0.04582798480987549,
-0.005440386012196541,
-0.05327755585312843,
-0.02288217470049858,
0.050602927803993225,
-0.001449938747100532,
-0.03511456400156021,
0.058663029223680496,
-0.035776231437921524,
-0.0012136290315538645,
-0.02368510141968727,
-0.021082885563373566,
-0.037486303597688675,
0.04487248882651329,
-0.01693914830684662,
0.024413680657744408,
-0.0040850224904716015,
-0.07593850791454315,
-0.105194091796875,
0.015171810984611511,
0.0047545041888952255,
0.09224969148635864,
-0.057417284697294235,
0.12594474852085114,
-0.03683807700872421,
-0.01664559170603752,
0.04238508641719818,
0.10188371688127518,
0.01873464696109295,
0.06933875381946564,
0.011076846159994602,
-0.01991232857108116,
-0.018443845212459564,
-0.05205276608467102,
0.048478107899427414,
0.008505796082317829,
0.009984740987420082,
0.020602302625775337,
-0.04838073253631592,
1.190894149664105e-31,
-0.052863650023937225,
-0.05952977016568184,
-0.05339684337377548,
0.055379122495651245,
0.0028937351889908314,
0.07006464898586273,
0.03316992148756981,
-0.033729903399944305,
0.005097357556223869,
0.016296641901135445,
0.003207416506484151,
0.021592313423752785,
-0.024926157668232918,
0.04623030871152878,
0.0626756101846695,
0.051180414855480194,
-0.025574391707777977,
-0.0039002024568617344,
-0.008390864357352257,
0.05272310599684715,
0.13916251063346863,
0.0183595959097147,
-0.014156167395412922,
0.027263429015874863,
-0.052319955080747604,
-0.0635838434100151,
-0.015033078379929066,
0.008359757252037525,
-0.08376339823007584,
-0.002965368330478668,
-0.01725870743393898,
0.01437150128185749,
0.02377946302294731,
-0.03852981701493263,
0.006717040669173002,
-0.06811609864234924,
0.09456215053796768,
0.03610585629940033,
-0.0001950231526279822,
0.06000695005059242,
-0.005336463451385498,
-0.10075848549604416,
0.0004325399058870971,
0.02728228084743023,
0.0917106494307518,
0.010467111133038998,
0.004023470915853977,
0.040424082428216934,
-0.048037514090538025,
0.022463539615273476,
0.07384366542100906,
-0.04513036832213402,
-0.014431679621338844,
-0.048047617077827454,
0.05496544390916824,
-0.00865122675895691,
0.07701002061367035,
0.030767066404223442,
-0.08740606158971786,
0.02144656702876091,
-0.008400587365031242,
0.10233189910650253,
0.025802040472626686,
0.0636770948767662
] |
1
Algebraic expressions
After completing this chapter you should be able to:
β Multiply and divide integer po
wers β pages 2β3
β Expand a single term over brackets and collect like
terms
β pages 3β4
β Expand the product of two or three expressions β pages 4β6
β Factorise linear, quadratic and simple cubic expressions β pages 6β9
β Know and use the laws of indices β pages 9β11
β Simplify and use the rules of surds β pages 12β13
β Rationalise denominators β pages 13β16Objectives
1 Simplify:
a 4m2n + 5mn2 β 2m2n + mn2 β 3mn2
b 3x2 β 5x + 2 + 3x2 β 7x β 12
β GCSE Mathematics
2 Write as a single power of 2:a
25 Γ 23 b 26 Γ· 22
c (23)2 β GCSE Mathematics
3 Expand:a
3(x
+ 4) b 5(2 β 3
x)
c 6(2x
β 5y) β GCSE Mathematics
4 Write down the highest common factor of:a
24 and 16 b 6x
and 8x2
c 4xy2 and 3xy β GCSE Mathematics
5 Simplify:
a 10x ____ 5 b 20x ____ 2 c 40x ____ 24
β GCSE MathematicsPrior knowledge check
Computer scientists use indices to describe
very large numbers. A quantum computer with 1000 qubits (quantum bits) can consider 2
1000
values simultaneously. This is greater than the number of particles in the observable universe.1
|
[
-0.08032307773828506,
0.06961223483085632,
-0.1076408326625824,
-0.015797344967722893,
0.006878102198243141,
0.0002554396050982177,
-0.05140487104654312,
0.07313896715641022,
-0.09630545228719711,
0.006601764354854822,
0.043856337666511536,
-0.08848366141319275,
-0.06750431656837463,
0.019747084006667137,
0.03018764778971672,
0.005788514856249094,
-0.06452768296003342,
0.136128768324852,
-0.03661727160215378,
-0.020722396671772003,
0.058406051248311996,
-0.019171619787812233,
-0.0569949708878994,
0.016241533681750298,
0.04578536003828049,
0.002280529821291566,
-0.007794762961566448,
0.002338041551411152,
0.057296205312013626,
-0.024352476000785828,
0.0497407503426075,
0.01957688108086586,
0.16482071578502655,
0.015752557665109634,
-0.03679561987519264,
-0.011297348886728287,
0.11618082225322723,
-0.023589355871081352,
-0.04679902642965317,
-0.015282935462892056,
-0.04405942186713219,
0.05097582936286926,
-0.012190804816782475,
0.015353099443018436,
0.0505477711558342,
-0.10876069962978363,
-0.03532164543867111,
-0.05329752340912819,
0.021127302199602127,
-0.06662769615650177,
-0.00815956387668848,
0.030114667490124702,
-0.019659075886011124,
-0.008133168332278728,
-0.03605879098176956,
-0.06578486412763596,
-0.024547886103391647,
0.03377404063940048,
-0.021862076595425606,
0.0034936415031552315,
-0.030726127326488495,
-0.009553461335599422,
0.021824102848768234,
0.004886041861027479,
0.016400957480072975,
0.05959289148449898,
-0.02409299835562706,
0.04318975657224655,
0.10211462527513504,
0.039984624832868576,
-0.07695768773555756,
-0.03585931658744812,
-0.06785189360380173,
-0.01731123961508274,
-0.0037342687137424946,
0.014159818179905415,
-0.060697041451931,
-0.05286484584212303,
-0.04599100723862648,
0.01752203330397606,
-0.027894984930753708,
0.1054912731051445,
0.09051688760519028,
-0.03333335742354393,
0.05526043847203255,
0.02516081929206848,
0.0744502916932106,
0.031163783743977547,
0.03749418631196022,
-0.06277690827846527,
0.08417759090662003,
-0.11109331250190735,
0.002563969697803259,
-0.06449735164642334,
-0.019047893583774567,
0.003677890170365572,
-0.004542990121990442,
-0.07659941166639328,
0.08767460286617279,
0.06912896782159805,
0.08231953531503677,
-0.045188747346401215,
0.025551091879606247,
-0.12346037477254868,
-0.0801108181476593,
-0.032263755798339844,
0.08946770429611206,
-0.04747356101870537,
0.034499023109674454,
-0.0984102413058281,
-0.09551644325256348,
0.04047062620520592,
-0.02472478337585926,
-0.007319318596273661,
0.05932087078690529,
-0.04369203373789787,
0.0686553418636322,
-0.03785654157400131,
0.04708113893866539,
0.002423602622002363,
0.06153596192598343,
-0.053143225610256195,
0.03225872293114662,
0.03863230347633362,
0.05413886532187462,
0.02769932523369789,
0.01967575028538704,
-0.0012061792658641934,
-0.011164950206875801,
0.01255554798990488,
0.04562809690833092,
0.1273660957813263,
-0.04785860329866409,
0.025827961042523384,
0.003028036328032613,
-0.06410640478134155,
-0.05221375823020935,
0.05925051122903824,
-0.041677746921777725,
0.008489580824971199,
-0.03862820938229561,
0.04677605628967285,
-0.011221133172512054,
-0.05522490665316582,
-0.01772642694413662,
-0.014810082502663136,
0.04987895488739014,
0.010693230666220188,
0.03433956205844879,
-0.03939329460263252,
-0.03387568145990372,
0.012060699984431267,
-0.017506256699562073,
0.046140171587467194,
0.04081380367279053,
0.0021655817981809378,
0.04947882518172264,
-0.06263557821512222,
0.0731547474861145,
-0.011616489849984646,
-0.030384229496121407,
-0.050322454422712326,
-0.042412471026182175,
0.044922877103090286,
-0.005648311693221331,
0.012453239411115646,
0.01949930563569069,
0.014309690333902836,
-0.08039113134145737,
0.04398651793599129,
0.013139096088707447,
-0.02753477729856968,
0.028544675558805466,
0.11865152418613434,
0.03379325196146965,
-0.032221872359514236,
0.07732171565294266,
-0.033725786954164505,
-0.035958439111709595,
-0.014703852124512196,
0.00579559151083231,
-0.039442017674446106,
-0.008187486790120602,
-0.004453151952475309,
0.034021034836769104,
-0.008257918059825897,
-0.04737914353609085,
0.05906597524881363,
0.0056229280307888985,
0.01713251695036888,
0.04135168716311455,
-0.03448183089494705,
-0.05333661660552025,
-0.032895904034376144,
-0.07447075843811035,
-0.005271414294838905,
0.06327188014984131,
-0.037583526223897934,
-0.09340319037437439,
0.007916387170553207,
-0.020627275109291077,
-0.048265863209962845,
-0.029557203873991966,
-0.05239401385188103,
0.05925383418798447,
-0.03787819296121597,
-0.07190479338169098,
-0.07056626677513123,
-0.0009868219494819641,
0.09991246461868286,
0.013306164182722569,
-0.01481883879750967,
0.04403619095683098,
-0.02945571392774582,
0.05798836797475815,
-0.01881047524511814,
0.05815602093935013,
0.018232228234410286,
-0.09673189371824265,
-0.023886071518063545,
-0.03590824082493782,
-0.057882972061634064,
0.006278167013078928,
-6.571407946388082e-33,
-0.06669445335865021,
-0.05881759896874428,
-0.09581144899129868,
-0.06676139682531357,
-0.04140564054250717,
-0.0077249426394701,
-0.008677075617015362,
-0.08284932374954224,
0.12371271848678589,
-0.002587558003142476,
-0.006817429792135954,
0.03454585373401642,
0.0343523733317852,
0.041253168135881424,
-0.046656787395477295,
-0.046103157103061676,
0.024066932499408722,
0.007944624871015549,
-0.003388836281374097,
-0.021050790324807167,
0.005544749554246664,
0.04707490652799606,
0.014038782566785812,
-0.023460185155272484,
-0.02893182635307312,
0.024006905034184456,
0.03407100960612297,
-0.04169299080967903,
0.0056815845891833305,
0.06662425398826599,
0.014873689040541649,
0.002268408425152302,
0.021725818514823914,
-0.034517012536525726,
0.026994038373231888,
-0.08545070141553879,
0.07845207303762436,
-0.015844685956835747,
-0.03320727497339249,
-0.015994098037481308,
0.10633629560470581,
-0.015725675970315933,
0.09519720077514648,
-0.07376612722873688,
0.012161562219262123,
0.01196275930851698,
0.0350465290248394,
-0.006787003017961979,
-0.020180614665150642,
-0.09955482184886932,
-0.02579047530889511,
-0.01729939877986908,
-0.0314486026763916,
0.030174752697348595,
0.04482046142220497,
-0.023791003972291946,
0.06383867561817169,
-0.06066443771123886,
0.06938644498586655,
0.026488937437534332,
-0.12063895165920258,
0.0003664848627522588,
0.019942045211791992,
-0.01898491196334362,
-0.025102972984313965,
-0.03145090118050575,
-0.017234262079000473,
-0.027514271438121796,
-0.043524887412786484,
0.0373549684882164,
-0.03605654835700989,
0.1077202782034874,
-0.00020353059517219663,
0.008311109617352486,
-0.009997735731303692,
-0.08986572176218033,
-0.012040800414979458,
-0.0164472758769989,
-0.015183234587311745,
-0.04987666383385658,
-0.02239920198917389,
-0.027397310361266136,
-0.03107280470430851,
-0.11525214463472366,
-0.13476289808750153,
0.036854662001132965,
0.06303198635578156,
0.0649033933877945,
-0.02143744006752968,
-0.05812320113182068,
-0.017744340002536774,
-0.027831312268972397,
0.06569142639636993,
-0.07001755386590958,
0.012766385450959206,
8.483151290265625e-32,
0.009790048003196716,
0.027932778000831604,
-0.04612995311617851,
-0.0165970791131258,
0.030833004042506218,
0.028729556128382683,
0.0280450489372015,
0.07328752428293228,
0.09445234388113022,
-0.01764804497361183,
0.06072142347693443,
0.05058182775974274,
-0.002746664686128497,
0.027960646897554398,
-0.07725425064563751,
-0.038846246898174286,
0.03276636451482773,
0.04960816726088524,
-0.04107784107327461,
0.004990806803107262,
-0.0008829428697936237,
0.030890407040715218,
0.006578345783054829,
0.0073388065211474895,
0.06667588651180267,
0.0986643061041832,
-0.07574061304330826,
0.02513548918068409,
-0.04980526119470596,
0.003930689767003059,
0.09934882074594498,
-0.0037609159480780363,
0.020206118002533913,
-0.009284631349146366,
0.02259337343275547,
0.0028158705681562424,
0.07383301109075546,
0.058942876756191254,
0.011773628182709217,
0.06736640632152557,
-0.09755080938339233,
-0.031024552881717682,
-0.05525270476937294,
0.00839043129235506,
0.0371619388461113,
-0.06424860656261444,
-0.08454108238220215,
-0.02135450392961502,
0.025599325075745583,
-0.003339921124279499,
-0.062303926795721054,
-0.07003919035196304,
0.008732268586754799,
-0.08134530484676361,
0.08742844313383102,
-0.04233838990330696,
-0.0009397914982400835,
-0.005168698262423277,
0.016924235969781876,
0.01848509907722473,
-0.015516799874603748,
0.09339199960231781,
-0.11654115468263626,
0.02781183458864689
] |
2
Chapter 1
1.1 Index laws
β You can use the laws of indices to simplify powers of the same base.
β’ am Γ an = am + n
β’ am Γ· an = am β n
β’ (am)n = amn
β’ (ab)n = anbn
Example 1
Example 2
Expand these expressions and simplify if possible:
a β3x
(7x β 4) b y2(3 β 2y3)
c 4x
(3x β 2x2 + 5x3) d 2x (5x + 3) β 5(2x + 3)Simplify these expressions:a
x2 Γ x5 b 2r2 Γ 3r3 c b7
__ b4 d 6x5 Γ· 3x3 e (a3)2 Γ 2a2 f (3x2)3 Γ· x4
x5Notation
This is the base .
This is the index, power or
exponent.
a x2Β ΓΒ x5 =Β x2 + 5 =Β x7
b 2r2Β ΓΒ 3 r3 =Β 2Β Γ Β 3Β ΓΒ r2Β ΓΒ r3
=Β 6Β ΓΒ r2 + 3 =Β 6 r5
c b7 __ b4 = b7 β 4 =Β b3
d 6x5Β Γ·Β 3 x3 =Β 6 __ 3 Β ΓΒ x5 ____ x3
=Β 2Β Γ
Β x2 = 2x2
e (a3)2Β ΓΒ 2 a2 =Β a6Β ΓΒ 2 a2
=Β 2Β ΓΒ a6Β ΓΒ a2 =Β 2 a8
f (3x2)3 _____ x4 =Β 33 Γ (x2)3 ____ x4
= 27
Γ x6 __ x4 =Β 27 x2Use the rule amΒ ΓΒ anΒ =Β am + n to simplify the index.
x5Β Γ·Β x3Β =Β x5 β 3Β =Β x2Rewrite the expression with the numbers
together and the r terms together.
2Β ΓΒ 3Β =Β 6
r2Β ΓΒ r3Β =Β r2 + 3
A min us sign outside
brackets changes the sign of every term inside the brackets.Watch outUse the rule am Γ· an = am β n to simplify the index.
Use the rule (am)nΒ =Β amn to simplify the index.
a6Β Γ a2 = a6 + 2 = a8
Use the rule (ab)nΒ =Β anbn to simplify the numerator.
(x2)3Β =Β x2 Γ 3Β =Β x6
x6 __ x4 Β =Β x6 β 4Β =Β x2
|
[
-0.024355515837669373,
0.08441449701786041,
0.021609827876091003,
-0.03640357032418251,
-0.03166563808917999,
0.014180147089064121,
-0.04340866208076477,
0.0018016294343397021,
0.022351857274770737,
0.017754772678017616,
-0.022316208109259605,
-0.03964490815997124,
0.06053663045167923,
-0.030213607475161552,
0.0217477735131979,
0.024524377658963203,
-0.022755393758416176,
0.16182611882686615,
-0.07453668117523193,
-0.013809720985591412,
0.09938043355941772,
-0.029093163087964058,
-0.10365865379571915,
0.0614762045443058,
0.008742669597268105,
0.003086949000135064,
-0.025924306362867355,
0.08538170903921127,
0.03455553576350212,
-0.13855108618736267,
0.057275231927633286,
0.0259725209325552,
0.08800867199897766,
-0.00322341057471931,
-0.013400506228208542,
-0.02553357556462288,
0.04368174448609352,
0.0665382519364357,
0.015682578086853027,
0.046493642032146454,
0.0458793044090271,
0.10816726833581924,
0.09638366848230362,
0.03858908638358116,
-0.03636092692613602,
-0.017326395958662033,
-0.05702987313270569,
-0.005662358831614256,
-0.04518857225775719,
-0.02457207441329956,
0.047993700951337814,
-0.03383606672286987,
-0.03358493372797966,
-0.013079800643026829,
0.0003265385748818517,
-0.021469030529260635,
0.008556247688829899,
0.02929525077342987,
-0.037230655550956726,
0.026135563850402832,
0.058589618653059006,
0.014820827171206474,
-0.0044006844982504845,
-0.006986403837800026,
-0.00030480168061330914,
0.01166801992803812,
0.047704484313726425,
-0.040509823709726334,
0.0064056022092700005,
0.026498647406697273,
-0.050007373094558716,
0.00127247697673738,
-0.014186845161020756,
-0.05312879756093025,
0.05211827903985977,
-0.003634708933532238,
-0.07154824584722519,
-0.03691895306110382,
0.01800381951034069,
0.11041805148124695,
-0.08963476121425629,
-0.02914491854608059,
0.07090426236391068,
0.010529916733503342,
0.055573493242263794,
-0.06423267722129822,
0.10628523677587509,
0.0070669339038431644,
0.10572673380374908,
-0.0010340657318010926,
-0.001496962970122695,
0.02478223294019699,
0.005931538995355368,
-0.04602242633700371,
0.009865058586001396,
0.013764344155788422,
0.0005611517699435353,
-0.04562614858150482,
0.039441727101802826,
0.0033858625683933496,
0.05241747573018074,
-0.12369094789028168,
-0.08287207037210464,
0.023957204073667526,
-0.06892411410808563,
-0.07052521407604218,
-0.008610108867287636,
-0.0753348097205162,
0.0897742286324501,
-0.030862348154187202,
-0.01849118061363697,
0.08788152039051056,
-0.02628154121339321,
-0.0015459131682291627,
0.01508621871471405,
-0.03942671790719032,
0.04242614284157753,
-0.020402543246746063,
0.07644930481910706,
-0.019139790907502174,
-0.011502914130687714,
-0.07231626659631729,
0.027624905109405518,
0.00479544373229146,
0.01721891388297081,
0.02744266949594021,
-0.0019268254982307553,
0.005371978040784597,
-0.07591008394956589,
-0.03947948291897774,
0.00647639250382781,
0.03142695501446724,
-0.004448168445378542,
0.005880054086446762,
-0.0357387512922287,
-0.025595612823963165,
-0.07774636894464493,
0.04172389209270477,
0.03526906296610832,
0.09928083419799805,
-0.006379208993166685,
-0.04011095315217972,
0.03329876437783241,
0.04182395711541176,
0.03171916306018829,
0.019208820536732674,
-0.021925063803792,
-0.019678646698594093,
0.025688719004392624,
-0.014937077648937702,
-0.019070738926529884,
-0.03705355152487755,
-0.03341471776366234,
0.02276117354631424,
-0.029208103194832802,
-0.06326613575220108,
0.03922374173998833,
-0.05215096473693848,
-0.01615525409579277,
0.006049078889191151,
-0.04581070691347122,
-0.01807793602347374,
-0.004264526069164276,
0.08104754239320755,
0.04614272341132164,
-0.019111089408397675,
0.02397904545068741,
0.05884716659784317,
-0.04318007081747055,
0.025710992515087128,
0.021368185058236122,
-0.007460764609277248,
-0.020308375358581543,
0.09690706431865692,
-0.0568581148982048,
-0.055373869836330414,
0.04405040293931961,
0.02001390978693962,
-0.034429267048835754,
-0.00653846375644207,
0.014427593909204006,
-0.13284654915332794,
-0.0034502968192100525,
0.039082836359739304,
-0.0351865217089653,
-0.007149970158934593,
-0.058075446635484695,
0.03066488541662693,
-0.042030978947877884,
-0.033770184963941574,
0.02066347934305668,
-0.01569848507642746,
-0.09881965816020966,
0.01942365989089012,
-0.005259593948721886,
0.025086818262934685,
0.007805072236806154,
-0.01933366246521473,
-0.04727660492062569,
0.03993985801935196,
0.03430236130952835,
0.019839240238070488,
0.01299882959574461,
-0.06337404996156693,
0.00007348118379013613,
-0.07489994168281555,
-0.04587334394454956,
-0.029898449778556824,
-0.006214877124875784,
0.13114231824874878,
-0.037449080497026443,
0.04550790414214134,
0.05605714023113251,
-0.035444386303424835,
0.03892943263053894,
0.0012588158715516329,
0.016770794987678528,
-0.019904272630810738,
-0.019161531701683998,
-0.05235384777188301,
-0.06253160536289215,
-0.07006558775901794,
0.024581963196396828,
-1.5625479063496036e-32,
-0.01155250333249569,
-0.06545447558164597,
-0.07928211241960526,
-0.06260491907596588,
-0.06541432440280914,
-0.01898123510181904,
0.08641105145215988,
-0.14129948616027832,
0.020574312657117844,
-0.002083870116621256,
-0.016856390982866287,
-0.01449218112975359,
0.0623055063188076,
-0.03754214569926262,
-0.10813198238611221,
-0.0686870589852333,
-0.004729611799120903,
0.04958377406001091,
0.036139730364084244,
-0.06718985736370087,
-0.07075563073158264,
0.05088181793689728,
-0.024846602231264114,
0.015986820682883263,
-0.04031645134091377,
-0.037166494876146317,
-0.02545079030096531,
-0.046644426882267,
0.11275940388441086,
0.04844260215759277,
0.0374700203537941,
-0.030697830021381378,
0.08247396349906921,
0.09611658751964569,
0.005261505022644997,
-0.06622837483882904,
0.03321440517902374,
0.018086321651935577,
-0.006951555144041777,
-0.027217615395784378,
-0.0031773033551871777,
-0.02183602750301361,
0.12592725455760956,
-0.04962291941046715,
-0.04982391372323036,
-0.021496592089533806,
-0.01589699275791645,
0.08519020676612854,
-0.04337099567055702,
-0.02095026522874832,
-0.03402495011687279,
-0.12502992153167725,
-0.05064934119582176,
-0.0022931008134037256,
-0.03663802146911621,
-0.044011812657117844,
0.09880255162715912,
0.002484607743099332,
0.02668796107172966,
-0.019534405320882797,
-0.06082763522863388,
0.022168995812535286,
0.01244127657264471,
0.022092487663030624,
-0.04805018752813339,
-0.013867348432540894,
-0.0004436652234289795,
-0.05816563591361046,
0.020538445562124252,
-0.027546154335141182,
-0.0068971519358456135,
0.0421731099486351,
-0.12791553139686584,
0.00195774482563138,
0.0064600994810462,
-0.1004607230424881,
-0.031226014718413353,
0.10155001282691956,
-0.10400716960430145,
-0.06162528693675995,
0.04416166618466377,
0.028511540964245796,
-0.05106577277183533,
-0.05403602495789528,
-0.0794311910867691,
0.006894579157233238,
0.09339134395122528,
0.052192918956279755,
0.01556074433028698,
-0.0531056709587574,
-0.06178890913724899,
-0.04360540211200714,
0.10780047625303268,
-0.09671835601329803,
0.08342104405164719,
9.621778024015898e-32,
0.08470131456851959,
-0.019938556477427483,
0.02774871699512005,
-0.036607787013053894,
0.01291392371058464,
-0.04752720147371292,
0.03465936332941055,
0.020613474771380424,
0.04776629060506821,
-0.03383554145693779,
0.03946888819336891,
0.076411172747612,
0.01942090131342411,
0.019503526389598846,
-0.000920638965908438,
-0.01644899509847164,
-0.03675147518515587,
0.001511569949798286,
0.006559845991432667,
0.07650836557149887,
-0.07399321347475052,
0.014913832768797874,
0.03270040825009346,
0.040218811482191086,
0.07308389991521835,
0.005087778903543949,
0.0099794277921319,
0.010389686562120914,
-0.002455169567838311,
0.0027426443994045258,
0.10062035918235779,
0.02173234522342682,
-0.053191184997558594,
0.020291388034820557,
-0.029142554849386215,
0.07736214250326157,
0.04305558651685715,
0.046551674604415894,
-0.044649187475442886,
0.01096393819898367,
-0.018651513382792473,
-0.11907246708869934,
-0.07330188155174255,
-0.02854675054550171,
0.020127402618527412,
-0.10566927492618561,
-0.14421527087688446,
-0.1049065962433815,
0.0141221908852458,
-0.07730724662542343,
0.0035290797241032124,
-0.00729282246902585,
0.01324366219341755,
0.049149591475725174,
0.04038209468126297,
-0.03604762256145477,
-0.029443759471178055,
0.05044720694422722,
0.001376743079163134,
0.021852267906069756,
0.015613152645528316,
-0.01093594916164875,
-0.054792292416095734,
-0.014175927266478539
] |
3Algebraic expressions
a β3x(7xΒ β 4 ) =Β β21x2Β +Β 12 x
b y2(3Β βΒ 2y3) =Β 3 y2Β βΒ 2y5
c 4x(3xΒ βΒ 2 x2Β +Β 5 x3)
=Β 12 x2Β βΒ 8 x3Β +Β 20 x4
d 2x(5xΒ +Β 3 )Β βΒ 5(2 xΒ +Β 3)
=Β 10 x2Β +Β 6 xΒ βΒ 10 xΒ βΒ 15
=Β 10 x2Β βΒ 4 xΒ βΒ 15
a x7 + x4 _______ x3 = x7 ___ x3 + x4 ___ x3
=
x7 β 3Β + x4 β 3 = x4Β + x
b 3x2 β 6x5 __________ 2x = 3 x 2 ____ 2x β 6 x 5 ____ 2x
= 3 __ 2 x 2 β 1 β 3x5 β 1 = 3x ___ 2 β 3 x 4
c 20x7 + 15x3 ____________ 5x2 = 20 x 7 _____ 5 x 2 + 15 x 3 _____ 5 x 2
= 4x7 β 2 + 3 x3 β 2 = 4 x 5 + 3 xExample 3β3xΒ ΓΒ 7xΒ ξ΅Β β21x1 +Β 1Β ξ΅Β β21x2
β3xΒ ΓΒ (β4)Β ξ΅Β ξ±12x
Remember a minus sign outside the brackets
changes the signs within the brackets.
Simplify 6xΒ βΒ 10x to give β4x.
Simplify these expressions:
a x 7 + x 4 ______ x 3 b 3 x 2 β 6 x 5 ________ 2x c 20 x 7 + 15 x 3 __________ 5 x 2 y2Β ΓΒ (β2y3)Β ξ΅Β β2y2 +Β 3Β ξ΅Β β2y5
Divide each term of the numerator by x 3 .
x1 is the same as x.
Divide each term of the numerator by 2x.
Simplify each fraction:
3 x 2 ____ 2x = 3 __ 2 Γ x 2 ___ x = 3 __ 2 Γ x2 β 1
β 6 x 5 ____ 2x = β 6 __ 2 Γ x 5 ___ x = β3 Γ x5 β 1
Divide each term of the numerator by 5x2.
1 Simplify these expressions:
a x3 Γ x4 b 2x3 Γ 3x2 c k3
__ k2
d 4p3
___ 2p e 3x3 ___ 3x2 f (y2)5
g 10x5 Γ· 2x3 h ( p3)2 Γ· p4 i (2a3)2 Γ· 2a3
j 8p4 Γ· 4p3 k 2a4 Γ 3a5 l 21a3b7 ______ 7ab4
m 9x2 Γ 3(x2)3 n 3x3 Γ 2x2 Γ 4x6 o 7a 4 Γ (3a 4)2
p (4y 3)3 Γ· 2y3 q 2a3 Γ· 3a2 Γ 6a5 r 3a4 Γ 2a5 Γ a3Exercise 1A
|
[
-0.03926916420459747,
0.02947307750582695,
0.05890563875436783,
-0.07547527551651001,
0.002689734101295471,
0.06421772390604019,
-0.0007130759768188,
-0.06890697777271271,
-0.04190916568040848,
0.03731192648410797,
-0.008234048262238503,
-0.10437041521072388,
-0.0252936240285635,
-0.03860585764050484,
0.047998782247304916,
-0.008133879862725735,
-0.016392139717936516,
0.08111535757780075,
-0.05695648491382599,
0.0017761023482307792,
0.05415651202201843,
-0.07298538833856583,
-0.03944544121623039,
0.05706111714243889,
0.04586000367999077,
0.008612851612269878,
-0.0050317770801484585,
0.02068895846605301,
-0.04951230064034462,
-0.011864462867379189,
-0.01430758647620678,
0.021910374984145164,
0.1148357093334198,
-0.0885014533996582,
0.10667344182729721,
0.0371878407895565,
0.024868804961442947,
0.04646513611078262,
-0.0497279055416584,
0.014095131307840347,
-0.0016315000830218196,
-0.009168475866317749,
0.04978315904736519,
0.035908907651901245,
0.007408083416521549,
-0.05766137316823006,
-0.0019330872455611825,
-0.01768970675766468,
0.006199554540216923,
-0.0649619922041893,
-0.008640325628221035,
0.006201400421559811,
-0.019580405205488205,
0.022717852145433426,
0.027370348572731018,
-0.060806214809417725,
0.05293215066194534,
0.05658760294318199,
-0.06322827190160751,
-0.04395850747823715,
-0.05324511229991913,
-0.01007002592086792,
-0.0006160563207231462,
0.01810557395219803,
-0.025173500180244446,
0.05624743178486824,
-0.010318615473806858,
-0.00647413544356823,
-0.02190018631517887,
0.028400110080838203,
-0.07217776030302048,
0.019253414124250412,
-0.024603964760899544,
-0.07948844134807587,
0.07946086674928665,
0.007362943142652512,
0.0038180318661034107,
-0.05909670144319534,
-0.0017285908106714487,
-0.029707521200180054,
-0.01903635822236538,
0.09207691997289658,
0.12410019338130951,
0.00044037430780008435,
0.02979365549981594,
-0.020509403198957443,
0.03754142299294472,
0.017004743218421936,
0.0005685579963028431,
0.03157329186797142,
-0.0021893512457609177,
-0.023510217666625977,
-0.03212155029177666,
-0.06377960741519928,
-0.03515039011836052,
-0.0819561704993248,
0.045109398663043976,
-0.10623406618833542,
0.053761184215545654,
0.06202974542975426,
0.07288580387830734,
-0.02325741946697235,
-0.029674429446458817,
-0.011689879931509495,
-0.04522189125418663,
-0.07054691016674042,
0.026614397764205933,
-0.06294118613004684,
0.03815996274352074,
-0.09488707035779953,
-0.04702235013246536,
-0.047541577368974686,
0.005338048096746206,
0.005565590225160122,
-0.05337439849972725,
0.023258360102772713,
0.03855916112661362,
-0.016560543328523636,
0.04209131374955177,
-0.09216869622468948,
-0.026303470134735107,
-0.048836443573236465,
0.09995506703853607,
0.05096302181482315,
0.054094232618808746,
-0.011689707636833191,
-0.0067113847471773624,
0.04066905751824379,
-0.10353794693946838,
-0.03131028264760971,
-0.013066191226243973,
-0.010769957676529884,
-0.06238853931427002,
-0.03416014090180397,
0.004344690125435591,
-0.07252667844295502,
-0.08048141002655029,
0.058205436915159225,
-0.032285500317811966,
-0.006094229873269796,
0.003868674859404564,
0.06669987738132477,
0.05453622341156006,
0.0677558183670044,
-0.007522066589444876,
0.009831094183027744,
-0.09029019623994827,
0.011569865047931671,
-0.0169008057564497,
0.014376362785696983,
-0.006630741525441408,
-0.015108590945601463,
-0.0606151707470417,
0.06350162625312805,
-0.034674957394599915,
-0.1197565495967865,
0.009172395803034306,
-0.1080130934715271,
0.055002808570861816,
0.03736726939678192,
0.029467720538377762,
-0.028534075245261192,
-0.026595348492264748,
0.10839836299419403,
0.04736800864338875,
0.03125996142625809,
0.05075946822762489,
0.09011796861886978,
-0.023530984297394753,
0.004440715536475182,
-0.0725269615650177,
-0.01684095337986946,
-0.03320862725377083,
0.04187528416514397,
0.021627508103847504,
0.005845544394105673,
0.07842423766851425,
-0.03944692760705948,
-0.07255671173334122,
0.07253623753786087,
0.07743547856807709,
-0.03973439708352089,
-0.06073034182190895,
0.01673988625407219,
-0.021834708750247955,
-0.00978162046521902,
-0.018744200468063354,
0.02692602574825287,
-0.031401459127664566,
-0.042050424963235855,
0.015228676609694958,
-0.09039377421140671,
-0.003063625656068325,
0.014600640162825584,
-0.06862474232912064,
0.05101650208234787,
-0.09623981267213821,
-0.01910003088414669,
-0.038924094289541245,
0.09797507524490356,
0.05736871063709259,
0.008336639031767845,
0.05044303461909294,
-0.04838033765554428,
-0.036483168601989746,
-0.021173961460590363,
-0.013519948348402977,
-0.058754391968250275,
0.019487712532281876,
0.08029904961585999,
-0.09098228812217712,
0.1123901978135109,
0.014886517077684402,
0.004350603558123112,
0.010626625269651413,
-0.05749358609318733,
0.08325611054897308,
-0.011842736043035984,
-0.0570465624332428,
-0.004886663053184748,
-0.07726465165615082,
-0.07385988533496857,
0.059760041534900665,
-1.0814819125949703e-32,
0.030756400898098946,
-0.01871706172823906,
-0.07573132961988449,
-0.12261343002319336,
-0.013575258664786816,
-0.008078537881374359,
0.024234237149357796,
-0.1210106909275055,
0.05504497140645981,
0.022128116339445114,
0.00905165821313858,
0.008494444191455841,
-0.014392007142305374,
-0.005852754693478346,
-0.007678750902414322,
-0.07879544049501419,
-0.050223641097545624,
-0.031806182116270065,
0.02074596658349037,
-0.05702899396419525,
0.0034869913943111897,
-0.0019958396442234516,
-0.07508883625268936,
0.02235822193324566,
-0.06353292614221573,
0.051951587200164795,
0.037902794778347015,
-0.05763886496424675,
0.05146323889493942,
0.033617909997701645,
-0.009399457834661007,
-0.03891465440392494,
0.06523887068033218,
0.0781259536743164,
0.053579140454530716,
-0.039584722369909286,
-0.032230667769908905,
0.031097091734409332,
0.013682892546057701,
-0.00702752685174346,
0.024361317977309227,
0.04812979698181152,
0.07875904440879822,
-0.002002139575779438,
-0.02869495563209057,
-0.009723777882754803,
-0.031216241419315338,
0.09603120386600494,
0.018491806462407112,
0.0676441490650177,
-0.054149407893419266,
-0.09838682413101196,
-0.09356031566858292,
0.01719534397125244,
0.0413464717566967,
0.002670339308679104,
0.0537969134747982,
-0.03818883001804352,
0.05962697044014931,
-0.05057164281606674,
-0.027652688324451447,
-0.08043299615383148,
0.07895106822252274,
-0.05768541246652603,
-0.011717233806848526,
0.07175514847040176,
-0.012015782296657562,
-0.06707628071308136,
-0.05542639642953873,
-0.01403890736401081,
0.020881133154034615,
0.0847875326871872,
-0.12508921325206757,
-0.0642799660563469,
-0.04308606684207916,
-0.026917755603790283,
-0.04334693029522896,
0.10272259265184402,
-0.028654219582676888,
-0.073989637196064,
-0.004901780281215906,
-0.0564933605492115,
0.0016956162871792912,
0.02060382440686226,
-0.08170667290687561,
-0.017481014132499695,
0.14449161291122437,
0.07864173501729965,
0.014526709914207458,
-0.006302475929260254,
0.026025066152215004,
-0.008953629992902279,
0.06695526838302612,
-0.0012645444367080927,
0.09352409094572067,
1.0308489481075595e-31,
-0.03190818428993225,
0.050291769206523895,
-0.059133823961019516,
-0.029132897034287453,
-0.004026516806334257,
-0.05189286917448044,
0.004285180475562811,
-0.018762126564979553,
0.045001935213804245,
-0.0996800884604454,
0.071991465985775,
0.07869203388690948,
-0.05972638726234436,
0.004122667480260134,
-0.03490656986832619,
0.022342968732118607,
-0.029932793229818344,
0.0196696650236845,
0.024560866877436638,
0.029504287987947464,
-0.07021203637123108,
0.06326185911893845,
-0.051865264773368835,
0.031106872484087944,
0.027050962671637535,
0.01854555308818817,
-0.02664869837462902,
-0.006720139179378748,
-0.04694482311606407,
-0.00033458363031968474,
0.09807971119880676,
0.030918288975954056,
-0.020969977602362633,
-0.050154950469732285,
0.046573247760534286,
0.08480854332447052,
0.04449838027358055,
0.029098395258188248,
0.04509710520505905,
0.01936452090740204,
0.10030177980661392,
-0.024707883596420288,
-0.00581473670899868,
-0.03659914433956146,
0.007517873775213957,
-0.10449506342411041,
-0.0674952045083046,
-0.06476856023073196,
0.024292171001434326,
-0.08491665124893188,
-0.04520656540989876,
0.0372033454477787,
0.02762112021446228,
-0.002174247521907091,
0.03208601474761963,
-0.0756097212433815,
0.012878095731139183,
0.02366742677986622,
-0.025805002078413963,
-0.02012203447520733,
0.0036456212401390076,
0.01720110885798931,
-0.04272063076496124,
0.037791457027196884
] |
4
Chapter 1
1.2 Expanding brackets
To find the product of two expressions you multiply each term in one expression by each term in the
other expression.
(x + 5)(4x β 2y + 3)x Γ
5 Γ= x(4x β 2y + 3) + 5(4x β 2y + 3)= 4x
2 β 2xy + 3x + 20x β 10y + 15
= 4x2 β 2xy + 23x β 10y + 15Multiplying each of the 2 terms in the first expression by each of the
3 terms in the second expression gives 2 Γ 3 = 6 terms.
Simplify your answer by collecting like terms.2 Expand and simplify if possible:
a 9(x β
2) b x(x
+ 9) c β3y
(4 β 3y)
d x(y
+ 5) e βx(3
x + 5) f β5x
(4x + 1)
g (4x
+ 5)x h β3y
(5 β 2y2) i β2x (5x β 4)
j (3x β
5)x2 k 3(x + 2) + (x β 7) l 5x β 6 β (3x β 2)
m 4(c +
3d 2) β 3(2c + d 2) n (r2 + 3t2 + 9) β (2r2 + 3t2 β 4)
o x(3x2 β 2x + 5) p 7y2(2 β 5y + 3y2) q β2y2(5 β 7y + 3y2)
r 7(x β
2) + 3(x + 4) β 6(x β 2) s 5x β
3(4 β 2x) + 6
t 3x2 β x(3 β 4x) + 7 u 4x( x + 3) β 2x(3x β 7) v 3x2(2x + 1) β 5x2(3x β 4)
3 Simplify these fractions:
a 6 x 4 + 10 x 6 _________ 2x b 3 x 5 β x 7 _______ x c 2 x 4 β 4 x 2 ________ 4x
d 8 x 3 + 5x ________ 2x e 7 x 7 + 5 x 2 ________ 5x f 9 x 5 β 5 x 3 ________ 3x
a (x + 5)(x + 2)
= x2 + 2x + 5 x + 10
=
x2 + 7x + 10
b (x
β 2y)(x2 + 1)
= x3 + x β 2x2y β 2 yExample 4
Expand these expressions and simplify if possible:
a (x
+ 5)(x + 2) b (x
β 2y)(x2 + 1) c (x β y)2 d (x + y)(3x β 2y β 4)
Multiply x by (x + 2) and then multiply 5 by (x + 2).
Simplify your answer by collecting like terms.
β2y Γ x2 = β2x2y
There are no like terms to collect.
|
[
-0.06958398222923279,
0.04130092263221741,
0.014117943122982979,
-0.027454771101474762,
-0.016314055770635605,
0.07407425343990326,
0.05066818743944168,
-0.041329819709062576,
-0.00023953552590683103,
0.06338217109441757,
-0.0230973232537508,
-0.06870998442173004,
0.01709219068288803,
0.00438944436609745,
-0.003692110301926732,
-0.006350250449031591,
-0.03896801918745041,
0.068571075797081,
-0.07532839477062225,
-0.013104451820254326,
0.10006146878004074,
-0.022417915984988213,
-0.03980335220694542,
-0.006348673719912767,
0.03206590190529823,
-0.007295501418411732,
-0.03607826679944992,
-0.05170546844601631,
0.026057235896587372,
-0.06013723090291023,
0.07185913622379303,
-0.015059316530823708,
0.05855674296617508,
-0.05135418102145195,
0.01672988571226597,
0.0009945349302142859,
0.04079873859882355,
0.019799239933490753,
0.01876603253185749,
-0.006505136843770742,
0.04064192622900009,
-0.003441069507971406,
-0.005232266150414944,
-0.03824319317936897,
-0.04793413355946541,
-0.07509376108646393,
-0.053225077688694,
-0.05117899179458618,
0.01778595522046089,
-0.05118979141116142,
-0.016079887747764587,
-0.01790374517440796,
-0.013481505215168,
0.007763228379189968,
-0.05111878737807274,
-0.010412356816232204,
-0.026172734797000885,
0.05785008892416954,
0.027269551530480385,
0.02922366000711918,
-0.04010055959224701,
-0.00205258559435606,
-0.026565374806523323,
0.05781776085495949,
-0.04769998416304588,
-0.008473457768559456,
-0.04938158020377159,
-0.049060169607400894,
0.012595811858773232,
0.009920299053192139,
-0.13374710083007812,
-0.08709700405597687,
-0.032208751887083054,
-0.007906638085842133,
0.09070754051208496,
0.07166711241006851,
-0.02434460259974003,
0.016920462250709534,
0.012944107875227928,
-0.07374803721904755,
0.02216983400285244,
0.06233806535601616,
0.08071628957986832,
-0.0019190860912203789,
0.058315347880125046,
-0.0816112831234932,
0.02568906545639038,
-0.04841873049736023,
0.05109409987926483,
-0.017790069803595543,
0.07093711942434311,
-0.0881885513663292,
-0.030876539647579193,
-0.03374221548438072,
0.07552552968263626,
-0.007974887266755104,
0.05480361357331276,
-0.08594575524330139,
0.08402556926012039,
0.0968719944357872,
0.021537503227591515,
-0.024284575134515762,
-0.05486061051487923,
-0.10151565819978714,
-0.09616018086671829,
-0.05147485435009003,
0.056132879108190536,
0.006534014828503132,
0.0703815445303917,
-0.0603364035487175,
-0.06006868556141853,
-0.017997102811932564,
0.02702438086271286,
0.006732474081218243,
-0.0109058553352952,
-0.08595137298107147,
0.12010149657726288,
-0.0033192012924700975,
-0.00339457206428051,
0.002314301673322916,
-0.011971628293395042,
-0.005015525035560131,
0.06400775164365768,
0.03384598344564438,
-0.0018804935971274972,
-0.0014876407803967595,
0.07297451049089432,
0.009348621591925621,
-0.00045153364771977067,
-0.0508357509970665,
-0.051978833973407745,
0.014362774789333344,
0.002189838094636798,
0.033368028700351715,
0.035686101764440536,
-0.08887235820293427,
-0.09075077623128891,
0.011688992381095886,
-0.06785649061203003,
-0.012521473690867424,
0.011263249441981316,
0.05421536788344383,
0.012040273286402225,
0.005893371067941189,
0.015420579351484776,
0.025675637647509575,
0.025522930547595024,
-0.030788391828536987,
0.061974138021469116,
0.07380389422178268,
0.0015126258367672563,
-0.05504845082759857,
-0.025892194360494614,
-0.026414142921566963,
0.05427895113825798,
-0.0651007741689682,
-0.04768983647227287,
-0.0409369133412838,
0.0269409641623497,
-0.0540178157389164,
-0.04309401661157608,
-0.04754478111863136,
-0.06828110665082932,
0.076850026845932,
0.002149539301171899,
-0.012964632362127304,
-0.01597175933420658,
0.07467393577098846,
-0.04679547995328903,
-0.007135595194995403,
0.028373096138238907,
0.030340438708662987,
-0.027532389387488365,
0.10002629458904266,
-0.014636478386819363,
0.030064357444643974,
0.025980880483984947,
-0.04232793301343918,
-0.08845329284667969,
0.07493141293525696,
0.038698721677064896,
-0.013993607833981514,
0.006697332486510277,
0.06544122099876404,
-0.013452878221869469,
-0.05715228244662285,
-0.1018906831741333,
-0.017400814220309258,
-0.1203632652759552,
0.07005090266466141,
0.07507193833589554,
-0.04436975717544556,
-0.028323018923401833,
0.0024666409008204937,
-0.03359497711062431,
0.02529691345989704,
0.01199344452470541,
-0.01571970246732235,
-0.04269074276089668,
0.009662216529250145,
0.023388836532831192,
-0.009524492546916008,
0.07749643921852112,
-0.02750215493142605,
-0.00036041572457179427,
-0.007307806517928839,
-0.03978179767727852,
-0.0524560920894146,
0.022533606737852097,
0.0745013952255249,
0.06291096657514572,
0.0541372112929821,
0.02701539173722267,
0.010266799479722977,
0.03526604548096657,
-0.019679682329297066,
0.07711769640445709,
-0.021977709606289864,
-0.06791150569915771,
0.02115461230278015,
-0.07677312940359116,
-0.060022126883268356,
0.044948142021894455,
-1.6722739857167285e-32,
-0.01432250626385212,
0.01955193653702736,
-0.01140526682138443,
-0.15932977199554443,
0.019752318039536476,
-0.11138874292373657,
-0.015686117112636566,
-0.1378507763147354,
0.09080956876277924,
0.1186431348323822,
-0.025571297854185104,
-0.039712466299533844,
-0.045786429196596146,
0.01619788259267807,
-0.08112427592277527,
-0.037120163440704346,
0.06796326488256454,
0.09300655871629715,
0.0031987738329917192,
-0.053731586784124374,
0.03383886069059372,
-0.05542315915226936,
0.0011317844036966562,
0.04642806202173233,
-0.033962029963731766,
0.01037588994950056,
0.09767194092273712,
-0.07161685824394226,
0.08430100977420807,
0.08290651440620422,
-0.0193472933024168,
-0.051025617867708206,
0.020982498303055763,
0.10370980203151703,
0.022343024611473083,
-0.06257496029138565,
-0.002380479359999299,
0.013789947144687176,
0.027274494990706444,
-0.06030121445655823,
0.04523639380931854,
-0.0013401005417108536,
0.0034736436791718006,
-0.0774519219994545,
0.0056642573326826096,
0.0035244568716734648,
0.035627298057079315,
0.05422637239098549,
-0.03870310261845589,
-0.06409526616334915,
-0.03000968135893345,
-0.07013234496116638,
-0.071229949593544,
-0.004499767441302538,
-0.04270923137664795,
0.021842459216713905,
0.11423417925834656,
-0.00037429199437610805,
0.04872043430805206,
0.018486175686120987,
-0.05014253035187721,
0.000902585918083787,
0.04307412728667259,
0.01904408633708954,
-0.009448811411857605,
0.03303030505776405,
0.007112292572855949,
-0.02914440631866455,
-0.0009759292006492615,
-0.028203949332237244,
-0.08602467179298401,
0.07642307132482529,
-0.057291530072689056,
-0.03347867354750633,
0.00801158882677555,
0.010230086743831635,
-0.07187991589307785,
0.04110999032855034,
-0.028851695358753204,
-0.062065575271844864,
0.005347185302525759,
-0.05182905122637749,
0.011402096599340439,
-0.02490290440618992,
-0.1680552214384079,
0.025846702978014946,
0.07429201900959015,
0.11834350228309631,
-0.045652493834495544,
-0.06576449424028397,
-0.06648830324411392,
-0.04773164168000221,
0.036324601620435715,
-0.08671139925718307,
0.03541140258312225,
9.839652963204932e-32,
0.01096189022064209,
-0.014009286649525166,
-0.01913098618388176,
-0.03858070820569992,
0.06269218027591705,
0.05753939598798752,
0.09563268721103668,
-0.01718391664326191,
0.09613063931465149,
-0.07973091304302216,
0.02387816272675991,
0.009176123887300491,
-0.028687719255685806,
0.06383965909481049,
-0.06822578608989716,
-0.05418634042143822,
-0.07761108130216599,
-0.08585168421268463,
-0.03407103195786476,
0.009015815332531929,
0.008586202748119831,
0.024891123175621033,
-0.02387867495417595,
-0.0003653154126368463,
-0.014328300952911377,
0.057204388082027435,
-0.039622653275728226,
-0.014582611620426178,
-0.02695184201002121,
0.0056365360505878925,
0.056652963161468506,
-0.050813186913728714,
-0.02944987453520298,
0.002458760514855385,
-0.01649562455713749,
0.030132096260786057,
0.09523285925388336,
0.09640568494796753,
0.010961569845676422,
0.06644529104232788,
-0.010039689019322395,
-0.06897818297147751,
0.037440571933984756,
0.01551778707653284,
-0.010157597251236439,
-0.018984677270054817,
-0.031586598604917526,
-0.0731772631406784,
0.017765240743756294,
-0.05436425283551216,
-0.046535830944776535,
0.10326892882585526,
0.04437749460339546,
0.013589146547019482,
-0.03320726379752159,
-0.030816340819001198,
0.02556172013282776,
-0.026703212410211563,
0.046203602105379105,
0.01477200910449028,
-0.05307415872812271,
-0.02172992192208767,
-0.025636976584792137,
-0.001793578965589404
] |
5Algebraic expressions
c (x β y)2
= (x β y)(x β y)
=
x2 β xy β xy + y2
= x2 β 2xy + y2
d (x + y)(3x β 2 y β 4)
= x(3x
β 2y β 4) + y (3x β 2 y β 4)
= 3x2 β 2xy β 4 x + 3 xy β 2 y2 β 4y
= 3x2 + xy β 4 x β 2 y2 β 4y
a x(2x + 3)(x β 7)
= (2x2 + 3 x)(x β 7)
= 2
x3 β 14 x2 + 3 x2 β 21x
= 2
x3 β 11 x2 β 21x
b x(5x β
3y)(2x β y + 4)
= (5x2 β 3 xy)(2x β y + 4)
= 5x2(2x β y + 4) β 3 xy(2x β y + 4)
= 10x3 β 5 x2y + 20 x2 β 6 x2y + 3 xy2
β 12 xy
= 10
x3 β 11 x2y + 20 x2 + 3 xy2 β 12 xy
c (x
β 4)( x + 3)( x + 1)
= (x2 β x β 12)( x + 1)
=
x2(x + 1) β x (x + 1) β 12( x + 1)
=
x3 + x2 β x2 β x β 12 x β 12
=
x3 β 13 x β 12Example 5
Expand these expressions and simplify if possible:
a x(2x
+ 3)(x β 7) b x(5x
β 3y)(2x β y + 4) c (x
β 4)(x + 3)(x + 1)
Be careful with minus signs. You need to change
every sign in the second pair of brackets when you multiply it out.
Choose one pair of brackets to expand first, for example:
(x β 4)(x + 3) = x
2 + 3x β 4x β 12
= x2 β x β 12
You multiplied together three linear terms, so the final answer contains an x
3 term.βxy β xy = β2xy
Multiply x by (3x β 2y β 4) and then multiply y by (3x β 2y β 4).
Start by expanding one pair of brackets:x(2x + 3) = 2x
2 + 3x
You could also have expanded the second pair of brackets first: (2x + 3)(x β 7) = 2x
2 β 11x β 21
Then multiply by x.(x β y)2 means (x β y) multiplied by itself.
1 Expand and simplify if possible:
a (x
+ 4)(x + 7) b (x
β 3)(x + 2) c (x
β 2)2
d (x β y)(2x + 3) e (x + 3y)(4x β y) f (2x β 4y)(3x + y)
g (2x
β 3)(x β 4) h (3x
+ 2y)2 i (2x + 8y)(2x + 3)
j (x
+ 5)(2x + 3y β 5) k (x
β 1)(3x β 4y β 5) l (x
β 4y)(2x + y + 5)
m (x
+ 2y β 1)(x + 3) n (2x
+ 2y + 3)(x + 6) o (4 β
y)(4y β x + 3)
p (4y
+ 5)(3x β y + 2) q (5y
β 2x + 3)(x β 4) r (4y
β x β 2)(5 β y)Exercise 1B
|
[
-0.02795177325606346,
0.04221849888563156,
0.028725972399115562,
-0.033842090517282486,
-0.08281520009040833,
0.0052421982400119305,
0.04036655277013779,
-0.0967983603477478,
-0.09548991173505783,
0.03975280374288559,
-0.03204657882452011,
-0.06335991621017456,
0.019244195893406868,
0.020372554659843445,
0.061943646520376205,
-0.01208943035453558,
-0.021637465804815292,
-0.015516878105700016,
-0.06827711313962936,
-0.008443827740848064,
0.0453142374753952,
-0.07202405482530594,
-0.010773126035928726,
0.05465054512023926,
0.041612084954977036,
-0.044888678938150406,
-0.011754974722862244,
-0.030329622328281403,
-0.0718381255865097,
-0.057028334587812424,
0.012927090749144554,
-0.024992067366838455,
0.04817445948719978,
-0.10216384381055832,
0.07379946112632751,
0.014701672829687595,
0.03079264424741268,
0.03635171800851822,
0.0018665054813027382,
-0.017139684408903122,
-0.03234277293086052,
-0.037733521312475204,
0.01173228770494461,
-0.003705365350469947,
-0.07422297447919846,
-0.024199802428483963,
0.02685028687119484,
-0.0041193547658622265,
0.04079621657729149,
-0.059011392295360565,
0.02478138729929924,
-0.000405531725846231,
-0.04743478447198868,
0.09430354088544846,
0.0678391084074974,
-0.045341573655605316,
0.03874150663614273,
0.023453431203961372,
0.03089885599911213,
0.00599948363378644,
-0.014637036249041557,
0.006468527484685183,
-0.02448759227991104,
0.05278586968779564,
-0.05054091289639473,
0.057738304138183594,
0.013435403816401958,
-0.04075797647237778,
0.04215686768293381,
-0.012899556197226048,
-0.03268548101186752,
-0.020026464015245438,
-0.04617491364479065,
-0.08511742204427719,
0.06549518555402756,
-0.03953282907605171,
-0.05052599310874939,
-0.044692181050777435,
0.10292606800794601,
-0.02261320687830448,
0.01129978895187378,
-0.006525157485157251,
0.06991047412157059,
0.004989824257791042,
0.01426219753921032,
-0.06025463715195656,
0.0019041268387809396,
0.04448391869664192,
0.0028472822159528732,
0.03490807116031647,
0.021904639899730682,
-0.02501937933266163,
-0.04694611579179764,
-0.07287009805440903,
-0.03941645100712776,
-0.14243395626544952,
0.12874868512153625,
-0.09498432278633118,
-0.013094204477965832,
0.09671687334775925,
0.057391077280044556,
0.022885762155056,
0.017715824767947197,
-0.08824925124645233,
-0.006047928240150213,
-0.061691779643297195,
0.08680743724107742,
-0.017889225855469704,
0.027110600844025612,
-0.05089804530143738,
-0.023682832717895508,
0.016360372304916382,
-0.00800289586186409,
0.029777875170111656,
-0.043872419744729996,
0.048587407916784286,
0.0864420086145401,
-0.030428068712353706,
0.08666356652975082,
-0.061948262155056,
-0.02290378324687481,
-0.08231171220541,
0.02715909667313099,
0.018385201692581177,
0.005686392076313496,
-0.05469715595245361,
-0.05595213919878006,
0.03841915726661682,
-0.0019275281811133027,
-0.07977570593357086,
0.0008825113764032722,
-0.007344833109527826,
-0.04867333173751831,
-0.040184833109378815,
-0.020846623927354813,
-0.04312996193766594,
-0.05905092507600784,
0.07072336971759796,
0.008044295944273472,
-0.03171893581748009,
0.014077316038310528,
0.053245242685079575,
0.07628399133682251,
0.03944301977753639,
-0.06644417345523834,
-0.009029197506606579,
-0.01678701676428318,
0.016657525673508644,
0.0033851605840027332,
0.07749993354082108,
0.02693023346364498,
-0.010613234713673592,
-0.057467274367809296,
0.02827487513422966,
0.0721714049577713,
-0.07321327924728394,
-0.014832546003162861,
-0.05838143453001976,
0.056339286267757416,
-0.041652072221040726,
0.028614269569516182,
-0.018309393897652626,
0.03267282247543335,
0.04488168656826019,
0.018524648621678352,
0.013177831657230854,
0.10851156711578369,
0.02675214782357216,
0.012170558795332909,
-0.012386251240968704,
-0.03948010876774788,
0.043625131249427795,
-0.01361656840890646,
0.024375544860959053,
0.011371382512152195,
-0.057689301669597626,
0.13436651229858398,
-0.00539119029417634,
-0.01912996731698513,
0.06845645606517792,
0.06003163382411003,
-0.07634956389665604,
-0.048562221229076385,
0.04184683784842491,
0.0007952689193189144,
-0.05564041808247566,
-0.0773732140660286,
0.012291363440454006,
-0.10861270874738693,
0.042148854583501816,
0.01308266818523407,
-0.11261915415525436,
-0.06822726875543594,
0.020241357386112213,
-0.09434419125318527,
0.05852096900343895,
-0.03190354257822037,
-0.021901054307818413,
-0.0649571344256401,
0.02163613773882389,
0.04867452010512352,
0.08337697386741638,
0.07970458269119263,
-0.04007384553551674,
0.02357851341366768,
-0.06716089695692062,
0.01209344994276762,
-0.13288365304470062,
0.006594034377485514,
0.07696495205163956,
0.05654319003224373,
0.0916474387049675,
0.04256635531783104,
0.05769828334450722,
0.03267783299088478,
-0.08287246525287628,
0.10625803470611572,
0.04321327060461044,
-0.05020298436284065,
-0.013081583194434643,
-0.06949418783187866,
-0.06858813017606735,
0.04163774475455284,
-1.5754446956828565e-32,
-0.007306346669793129,
-0.015030549839138985,
-0.06231042370200157,
-0.09640008956193924,
-0.009928535670042038,
-0.05232590436935425,
0.008366208523511887,
-0.07004418224096298,
0.05919786915183067,
0.06558361649513245,
0.022234246134757996,
0.04574992135167122,
-0.010015293024480343,
0.05745454505085945,
-0.0022167053539305925,
-0.06443024426698685,
-0.05265556275844574,
-0.014102610759437084,
-0.008849823847413063,
-0.022511526942253113,
0.04320906847715378,
0.0006780880503356457,
-0.02923939749598503,
0.03804364427924156,
-0.06122921407222748,
-0.022473812103271484,
0.040059588849544525,
-0.04542486369609833,
0.07983408868312836,
0.027471350505948067,
-0.04162677004933357,
-0.09746681153774261,
0.039512597024440765,
0.09595204144716263,
0.04162115231156349,
-0.061113085597753525,
-0.0003598307375796139,
-0.026680640876293182,
0.04746302217245102,
-0.016412148252129555,
-0.02677828073501587,
0.02143256366252899,
0.0561569407582283,
0.014922857284545898,
0.03274683654308319,
0.026937836781144142,
-0.004728065803647041,
0.11991199851036072,
-0.02363388054072857,
-0.04862084984779358,
-0.07322011888027191,
-0.10603346675634384,
-0.07435139268636703,
0.013855685479938984,
0.04566946253180504,
-0.035980939865112305,
0.07217536866664886,
-0.02861565537750721,
0.05010795220732689,
0.03344119340181351,
-0.03586893901228905,
-0.1033734530210495,
0.031033573672175407,
0.003755932906642556,
-0.006010482087731361,
-0.024072518572211266,
0.016215577721595764,
-0.051677923649549484,
-0.040402770042419434,
-0.025777852162718773,
0.04710889607667923,
0.06242213770747185,
-0.13070900738239288,
-0.04716194421052933,
-0.06103331968188286,
-0.048226334154605865,
-0.06386777758598328,
0.041658174246549606,
-0.01854102313518524,
0.018868831917643547,
-0.034461744129657745,
-0.0015652779256924987,
0.07476342469453812,
-0.010471808724105358,
-0.11153817921876907,
-0.02992875687777996,
0.07475250214338303,
0.06519213318824768,
-0.009786947630345821,
0.007582302670925856,
0.004014992155134678,
0.007057603448629379,
0.0565013661980629,
-0.08225833624601364,
0.07430907338857651,
1.148418013945447e-31,
-0.02170920930802822,
-0.012290936894714832,
-0.07279406487941742,
-0.04041281342506409,
-0.02489219419658184,
-0.013906137086451054,
0.03202669695019722,
-0.042350299656391144,
0.021692832931876183,
-0.124973826110363,
0.0007790258387103677,
0.08167421072721481,
-0.09530327469110489,
0.00756649998947978,
-0.03443027660250664,
0.029500750824809074,
-0.03374732658267021,
0.02296564355492592,
-0.05755101144313812,
0.02345914952456951,
-0.06504911184310913,
-0.03566645830869675,
-0.04864896833896637,
-0.06028961390256882,
-0.022620422765612602,
0.05309251695871353,
-0.03146276995539665,
-0.0067274076864123344,
-0.056177057325839996,
0.005668402183800936,
0.06718689948320389,
-0.020028257742524147,
-0.06376538425683975,
-0.03822990879416466,
0.06864244490861893,
0.04345531761646271,
-0.0161899346858263,
0.07747329026460648,
0.030381130054593086,
0.08149397373199463,
0.05301579833030701,
-0.047026582062244415,
0.0011118138208985329,
0.004140776116400957,
0.015438826754689217,
-0.03625757247209549,
-0.024299966171383858,
-0.03989434242248535,
0.01947697252035141,
-0.08291004598140717,
-0.049408432096242905,
0.03755206987261772,
0.02416917122900486,
0.052762314677238464,
-0.006454591639339924,
-0.03929707407951355,
0.051903460174798965,
0.06844200193881989,
-0.004716364201158285,
0.015151284635066986,
-0.0125300707295537,
0.05193937197327614,
-0.03175956383347511,
0.004924289416521788
] |
6
Chapter 1
1.3 Factorising
You can write expressions as a product of their factors.
β Factorising is the opposite of expanding
brack
ets.4x(2x + y)
(x + 5)3
(x + 2y)(x β 5y)= 8x2 + 4xy
= x3 + 15x2 + 75x + 125
= x2 β 3xy β 10y2Expanding brackets
FactorisingExpand and simplify ( x + y )4. You can use the binomial expansion to expand
ex
pressions like ( x + y )4 quickly. β Section 8.3LinksChallenge2 Expand and simplify if possible:
a 5(x
+ 1)(x β 4) b 7(x
β 2)(2x + 5) c 3(x
β 3)(x β 3)
d x(x
β y)(x + y) e x(2x
+ y)(3x + 4) f y(x
β 5)(x + 1)
g y(3x
β 2y)(4x + 2) h y(7 β
x)(2x β 5) i x(2x
+ y)(5x β 2)
j x(x
+ 2)(x + 3y β 4) k y(2x
+ y β 1)(x + 5) l y(3x
+ 2y β 3)(2x + 1)
m x(2x
+ 3)(x + y β 5) n 2x
(3x β 1)(4x β y β 3) o 3x
(x β 2y)(2x + 3y + 5)
p (x
+ 3)(x + 2)(x + 1) q (x
+ 2)(x β 4)(x + 3) r (x
+ 3)(x β 1)(x β 5)
s (x
β 5)(x β 4)(x β 3) t (2x
+ 1)(x β 2)(x + 1) u (2x
+ 3)(3x β 1)(x + 2)
v (3x
β 2)(2x + 1)(3x β 2) w (x
+ y)(x β y)(x β 1) x (2x
β 3y)3
3 The diagram shows a rectangle with a square cut out.
The rectangle has length 3
x β y + 4 and width x + 7.
The square has length x β 2.Find an expanded and simplified expression for the shaded area.
x β 2x + 7
3x β y + 4
4 A cuboid has dimensions x + 2 cm, 2x β 1 cm and 2x + 3 cm.
Show tha
t the volume of the cuboid is 4x3 + 12x2 + 5x β 6 cm3.
5 Given tha
t (2x + 5y)(3x β y)(2x + y) = ax3 + bx2y + cxy2 + dy3, where a, b, c and d are
constants, find the values of a, b, c and d. (2 marks)P
Use the same strategy as you would use
if the lengths were given as numbers:
3cm6cm
10cmProblem-solving
P
E/P
|
[
-0.040166813880205154,
0.04968535900115967,
0.07222872227430344,
0.014693714678287506,
-0.020470233634114265,
0.06292809545993805,
0.010992651805281639,
-0.07985684275627136,
0.007047095336019993,
0.024562876671552658,
-0.060797128826379776,
-0.06879173219203949,
0.0034833953250199556,
0.033548641949892044,
0.03190777078270912,
-0.006980429869145155,
0.0030779216904193163,
0.057862065732479095,
-0.10496816039085388,
-0.043935567140579224,
0.0451287180185318,
-0.04732797294855118,
-0.03507717698812485,
-0.02653377316892147,
0.006250332575291395,
0.04004655033349991,
-0.04174831137061119,
-0.06288108974695206,
0.007466841023415327,
-0.031855810433626175,
0.06690683960914612,
0.006474129855632782,
0.07236874848604202,
-0.08537624031305313,
0.008053957484662533,
0.035815946757793427,
0.04567306116223335,
0.03363676369190216,
-0.0037488562520593405,
0.050706516951322556,
0.022534718737006187,
-0.009726221673190594,
0.009753071703016758,
0.021959224715828896,
-0.012980268336832523,
-0.07431571930646896,
0.028155270963907242,
-0.020682698115706444,
-0.016936104744672775,
-0.07384051382541656,
0.01471144799143076,
-0.02802916429936886,
-0.05025629326701164,
-0.011898821219801903,
-0.00935348216444254,
0.013925875537097454,
0.012173885479569435,
0.07165475934743881,
-0.08045665919780731,
0.0046667675487697124,
0.008540595881640911,
0.01911991834640503,
-0.023096000775694847,
0.04740453511476517,
-0.000659387034829706,
-0.045561593025922775,
-0.024516895413398743,
-0.037633366882801056,
-0.005881198216229677,
0.04873903468251228,
-0.029459893703460693,
-0.07243023812770844,
0.006320424377918243,
0.015498962253332138,
0.057779278606176376,
0.08488274365663528,
-0.08911262452602386,
-0.007630533538758755,
0.03461841866374016,
-0.04162587225437164,
0.01430433988571167,
0.039009664207696915,
0.03996279835700989,
-0.02343408204615116,
0.05386936664581299,
-0.031158016994595528,
0.011596053838729858,
0.022755879908800125,
0.04626019299030304,
0.005137817468494177,
-0.0026198942214250565,
-0.06824428588151932,
-0.013355692848563194,
-0.06101140379905701,
0.043165385723114014,
-0.06610732525587082,
0.06935128569602966,
-0.04496253281831741,
0.01835605315864086,
0.023731840774416924,
0.050881560891866684,
0.010793004184961319,
-0.05438865348696709,
-0.13350893557071686,
-0.08861245214939117,
-0.006784654688090086,
0.015962472185492516,
0.016305014491081238,
0.04432915896177292,
-0.0630878210067749,
-0.128694549202919,
0.017476173117756844,
0.028118476271629333,
-0.047792334109544754,
-0.0697401762008667,
-0.06460031867027283,
0.07753245532512665,
-0.021574825048446655,
0.05764426290988922,
-0.03432021290063858,
-0.00964296143501997,
-0.015420697629451752,
0.009423627518117428,
0.010472340509295464,
0.02475997433066368,
-0.04386390000581741,
0.012373660691082478,
-0.009435216896235943,
0.01876298524439335,
-0.074711374938488,
0.03312939405441284,
0.039284005761146545,
-0.0013670994667336345,
0.0679013580083847,
-0.015259569510817528,
-0.1100514605641365,
-0.05558431148529053,
0.06769486516714096,
-0.012305326759815216,
0.024357782676815987,
0.04387972503900528,
0.07042856514453888,
0.08985856920480728,
0.013801660388708115,
-0.055333685129880905,
-0.011130396276712418,
0.028665857389569283,
0.011068310588598251,
0.045353177934885025,
0.010576104745268822,
-0.06915343552827835,
-0.05479927733540535,
-0.05581997334957123,
-0.003527239663526416,
0.05158982798457146,
-0.06392140686511993,
0.004378157667815685,
-0.04356536641716957,
-0.007812362629920244,
0.00652197515591979,
-0.024693094193935394,
0.0033053874503821135,
-0.09335940331220627,
0.037243474274873734,
-0.039211150258779526,
-0.0417010597884655,
0.025326116010546684,
0.10344574600458145,
-0.025966480374336243,
0.013984648510813713,
-0.005483701825141907,
0.026459241285920143,
-0.05567843094468117,
0.07774175703525543,
-0.023401722311973572,
0.001369214616715908,
0.03593522310256958,
-0.0668778046965599,
0.032762330025434494,
0.05160917714238167,
0.08237839490175247,
-0.03396371379494667,
0.016328884288668633,
0.07165925204753876,
-0.024950111284852028,
-0.07487384974956512,
-0.028288215398788452,
-0.05780533701181412,
-0.04890579357743263,
-0.035803209990262985,
0.04980381950736046,
-0.07270120084285736,
-0.07470648735761642,
0.026555662974715233,
0.04439333826303482,
0.06020817160606384,
-0.039222970604896545,
0.013387830927968025,
-0.026757484301924706,
0.022806428372859955,
0.041535552591085434,
0.08101878315210342,
0.08411423116922379,
-0.07397850602865219,
-0.0037898484151810408,
0.059466682374477386,
-0.020125100389122963,
-0.054755799472332,
0.05268879979848862,
0.1398221105337143,
0.05719715729355812,
0.04490234702825546,
0.03344176337122917,
0.06470049917697906,
-0.013924896717071533,
-0.0029827768448740244,
0.02975328266620636,
0.02895999327301979,
-0.0635979101061821,
-0.03641296178102493,
0.011761317029595375,
-0.08413949608802795,
0.0331183560192585,
-1.4055594603029363e-32,
-0.06442949920892715,
-0.03132282570004463,
-0.037943460047245026,
-0.0582101047039032,
-0.029650578275322914,
-0.1427004039287567,
0.035544708371162415,
-0.14040353894233704,
0.14444555342197418,
0.07316440343856812,
-0.03288983181118965,
-0.04911187291145325,
0.017369896173477173,
-0.008246351033449173,
-0.0668352022767067,
-0.04942779988050461,
-0.04476665332913399,
0.07092612236738205,
0.008201123215258121,
-0.0859956219792366,
0.015249721705913544,
-0.09402149170637131,
0.011439526453614235,
0.04819200932979584,
-0.07658680528402328,
-0.011073455214500427,
0.03865232318639755,
-0.02736470103263855,
0.08431888371706009,
0.07903195172548294,
-0.0021056535188108683,
-0.025688765570521355,
-0.0008070892654359341,
0.08292993903160095,
0.020876269787549973,
-0.08517814427614212,
0.018396727740764618,
0.0013423999771475792,
0.0730268657207489,
-0.022675171494483948,
0.03720178082585335,
-0.039471421390771866,
0.0003449228242971003,
-0.032826587557792664,
0.023899463936686516,
-0.011240716092288494,
0.016801554709672928,
0.02779792807996273,
-0.07314583659172058,
-0.041274845600128174,
-0.050561390817165375,
-0.08546458184719086,
-0.10024506598711014,
0.055701710283756256,
-0.08728070557117462,
-0.003772762604057789,
0.1400538831949234,
-0.021726349368691444,
0.014414461329579353,
-0.059354014694690704,
-0.05049677565693855,
-0.044779110699892044,
0.04894821718335152,
-0.0430847704410553,
-0.01727772317826748,
0.032774168998003006,
-0.027569536119699478,
0.009955385699868202,
-0.0014604692114517093,
-0.01427866518497467,
0.01792820729315281,
0.06411471217870712,
-0.06446229666471481,
0.057404834777116776,
0.0016056395834311843,
-0.050709981471300125,
-0.031729117035865784,
0.019027600064873695,
-0.04450010880827904,
-0.01717054471373558,
0.08231619745492935,
-0.06047862395644188,
0.010977517813444138,
-0.019501712173223495,
-0.13007138669490814,
0.010064061731100082,
0.08197642117738724,
0.08565234392881393,
-0.067923903465271,
-0.04167310148477554,
-0.07846126705408096,
-0.06440721452236176,
0.09223313629627228,
-0.0908660888671875,
0.016050847247242928,
1.0750904412422825e-31,
0.031080570071935654,
-0.013714928179979324,
-0.0331295020878315,
0.013776182197034359,
0.06501425057649612,
0.012355290353298187,
0.06800244003534317,
0.007064954377710819,
0.0554809533059597,
-0.12536165118217468,
0.020819855853915215,
0.024406222626566887,
0.033491529524326324,
0.013386795297265053,
-0.03834608942270279,
-0.0412154346704483,
-0.0007738726562820375,
-0.009286368265748024,
-0.08705170452594757,
0.021020054817199707,
-0.03685817867517471,
0.03193719685077667,
-0.008097844198346138,
0.05551948770880699,
-0.04423735290765762,
0.017509566619992256,
-0.0014474053168669343,
-0.03275328502058983,
-0.009692317806184292,
-0.027083223685622215,
0.042256128042936325,
-0.004751900676637888,
-0.04964777082204819,
-0.0053024934604763985,
-0.049722302705049515,
0.1194283589720726,
0.05860229581594467,
0.15286533534526825,
-0.06160963326692581,
0.06627313047647476,
-0.030363667756319046,
-0.13275974988937378,
0.03782713785767555,
0.021286245435476303,
-0.002532518468797207,
-0.06261369585990906,
-0.009886913932859898,
-0.03904180973768234,
0.011538498103618622,
-0.00964819174259901,
0.006331210024654865,
0.023386960849165916,
0.07202037423849106,
0.027451371774077415,
-0.05796551704406738,
-0.014576931484043598,
-0.012332885526120663,
0.00578264007344842,
0.006929451134055853,
0.006953265983611345,
-0.06608600169420242,
0.003819134784862399,
0.03833566978573799,
-0.041686251759529114
] |
7Algebraic expressions
An ex pression in the form x2 β y2 is
called the difference of two squares.Notation= (x + 3)(2x β 1)β A quadratic expression has the form
ax2 + bx + c where a, b and c are real
numbers and a β 0.
To factorise a quadratic expression:
β’Find two fact
ors of ac that add up to b
β’Rewrite the
b term as a sum of these two
factors
β’Factorise each p
air of terms
β’Take out the c
ommon factor
β x2 β y2 = (x + y)(x β y)a 3x + 9 = 3( x + 3)
b x2 β 5 x = x(x β 5)
c 8x2 + 20 x = 4 x(2x + 5)
d 9x2y + 15 xy2 = 3 xy(3x + 5 y)
e 3x2 β 9 xy = 3 x(x β 3 y)Example 6
Factorise these expressions completely:
a 3x
+ 9 b x2 β 5 x c 8x2 + 20x d 9x2y + 15xy2 e 3x2 β 9xy
3 is a common factor of 3x and 9.
For the expression 2x2 + 5x β 3, ac = β6 = β1 Γ 6
and β1 + 6 = 5 = b.
2x2 β x + 6x β 3
= x(2x β 1) + 3(2x β 1)x is a common factor of x2 and β5x.
4 and x are common factors of 8x2 and 20x.
So take 4x outside the brackets.
3, x and y are common factors of 9x2y and 15xy2.
So take 3xy outside the brackets.
x and β3y have no common factors so this
expression is completely factorised.
Real n umbers are all the positive and
negative numbers, or zero, including fractions and surds.Notation
Example 7
Factorise:
a x2Β βΒ 5xΒ βΒ 6 b x2Β +Β 6xΒ +Β 8 c 6x2Β βΒ 11xΒ βΒ 10 d x2Β βΒ 25 e 4x2Β βΒ 9y2
a x2 β 5 x β 6
ac =
β6 and b = β 5
So x2 β 5 x β 6 ξ΅ x2 + x β 6 x β 6
= x(x
+ 1) β 6( x + 1)
= (x
+ 1)( x β 6)Here aΒ =Β 1, bΒ =Β β 5 and cΒ =Β β 6.
1 Work out the two factors of ac =Β β 6 which add
t
o give you bΒ =Β β5. β6 + 1Β =Β β5
2 Rewrite the b term using these two factors.
3 Factorise first two terms and last two terms.
4 xΒ + 1 is a factor of both terms, so take that
outside the brackets. This is now completely
factorised.
|
[
-0.009509725496172905,
0.10108181089162827,
0.004270889796316624,
0.007214569952338934,
-0.037485163658857346,
0.0566447414457798,
-0.03139231353998184,
-0.07215012609958649,
-0.026578273624181747,
-0.004736719653010368,
-0.023380352184176445,
-0.04062366858124733,
0.0017067048465833068,
0.010336294770240784,
0.11363238096237183,
-0.0002849849406629801,
-0.031959861516952515,
-0.00044707191409543157,
-0.04606156796216965,
-0.043825335800647736,
0.11850715428590775,
-0.08920035511255264,
-0.0809294804930687,
0.025427190586924553,
0.05870790034532547,
0.016033882275223732,
0.005917149130254984,
-0.02841479331254959,
0.014430812560021877,
0.02315579541027546,
-0.0140268849208951,
0.0531696192920208,
0.06861697882413864,
-0.05558406561613083,
0.09885318577289581,
-0.03624904155731201,
0.08030768483877182,
-0.0033585582859814167,
-0.05154961720108986,
-0.03479614108800888,
-0.045447275042533875,
-0.015507116913795471,
-0.017666181549429893,
0.023817267268896103,
-0.0211408119648695,
-0.03208471089601517,
0.03249955549836159,
-0.04419422149658203,
0.05279798433184624,
-0.041875917464494705,
0.004677856806665659,
0.026060182601213455,
-0.10082466155290604,
0.020690742880105972,
-0.01140337623655796,
-0.04365915805101395,
-0.025180969387292862,
0.04547307267785072,
-0.02631218731403351,
-0.02050597406923771,
-0.021682899445295334,
-0.022591935470700264,
0.013390612788498402,
0.04673148691654205,
-0.0031828810460865498,
-0.009743637405335903,
-0.022717781364917755,
-0.014253851026296616,
0.05869331210851669,
0.04339078813791275,
-0.006380861159414053,
0.018755776807665825,
0.01635431870818138,
0.006988896988332272,
0.019097181037068367,
0.018385685980319977,
-0.07119040191173553,
-0.011886200867593288,
0.12755316495895386,
0.021480519324541092,
0.007834125310182571,
-0.017712794244289398,
0.051569242030382156,
0.02531815506517887,
-0.03603431582450867,
-0.061105500906705856,
0.06547100096940994,
0.002240738132968545,
0.015113556757569313,
-0.00936915259808302,
0.03809015452861786,
-0.01924845762550831,
-0.02262387052178383,
-0.07340183109045029,
-0.00306135811842978,
-0.09968080371618271,
0.04177785664796829,
-0.031460512429475784,
0.05757613852620125,
0.05675439536571503,
0.06925371289253235,
-0.03551715612411499,
-0.022591738030314445,
-0.03645734861493111,
-0.028222184628248215,
-0.04788314551115036,
0.021612565964460373,
-0.052785418927669525,
0.06815400719642639,
-0.03918317332863808,
-0.12226857244968414,
-0.032349593937397,
-0.016883619129657745,
0.026561034843325615,
-0.023644307628273964,
-0.020650187507271767,
0.1313454955816269,
-0.05031526833772659,
0.13879504799842834,
-0.104586161673069,
0.054973453283309937,
-0.05631311610341072,
0.1226511001586914,
-0.03744887188076973,
0.017107373103499413,
-0.013001766987144947,
-0.024190323427319527,
0.000571615993976593,
-0.020362095907330513,
-0.055281542241573334,
0.04778918996453285,
0.04842761158943176,
-0.032543715089559555,
-0.02939811535179615,
0.020801423117518425,
-0.0857437252998352,
-0.060687240213155746,
0.11233670264482498,
-0.023294884711503983,
0.007046244107186794,
0.10430409014225006,
0.007713305298238993,
0.029607439413666725,
0.05298876762390137,
-0.0559147484600544,
-0.04042039066553116,
0.03444257751107216,
-0.0033453735522925854,
-0.028786875307559967,
0.0311171505600214,
-0.02801928110420704,
0.01275731436908245,
-0.03948219493031502,
0.029671592637896538,
0.09292995184659958,
-0.06210993602871895,
0.031469061970710754,
-0.07946550846099854,
-0.00794075895100832,
-0.014504742808640003,
-0.05028948932886124,
-0.030534842982888222,
-0.050809867680072784,
0.029181692749261856,
0.027981873601675034,
-0.005666723940521479,
0.024893203750252724,
0.0556238554418087,
0.058704402297735214,
0.0307004451751709,
-0.043855369091033936,
0.05037405714392662,
-0.0022130999714136124,
-0.01262078620493412,
0.06525905430316925,
-0.04383933171629906,
0.047707922756671906,
-0.05300454795360565,
0.03549975901842117,
0.09914340823888779,
0.10499263554811478,
-0.020729906857013702,
-0.025978058576583862,
0.014261743985116482,
0.03147060051560402,
-0.08073557168245316,
-0.021723512560129166,
0.002085196552798152,
-0.01126227155327797,
0.0007842471241019666,
0.0542139895260334,
-0.07939383387565613,
-0.0782330334186554,
0.05624621734023094,
-0.05014219507575035,
0.008834263309836388,
-0.06104765087366104,
0.009803357534110546,
-0.13537625968456268,
0.013924832455813885,
0.05136383697390556,
0.07733302563428879,
0.031823836266994476,
-0.06146402657032013,
-0.020137516781687737,
-0.00963932741433382,
-0.0251846294850111,
-0.09005704522132874,
-0.04535236582159996,
0.033673614263534546,
-0.033153630793094635,
0.0466904416680336,
-0.007396115455776453,
-0.005507204681634903,
-0.0727696418762207,
-0.100423164665699,
0.004284638445824385,
0.009512349031865597,
-0.0656629279255867,
0.008398895151913166,
-0.03185921162366867,
-0.05192417651414871,
0.005984618328511715,
-5.718874423640479e-33,
-0.021353228017687798,
-0.016870994120836258,
-0.10223376005887985,
0.00260270107537508,
-0.034709032624959946,
-0.038751401007175446,
0.04175952821969986,
-0.1104755848646164,
0.12629029154777527,
0.06392449885606766,
-0.020167814567685127,
0.012149208225309849,
0.03433005511760712,
-0.061121322214603424,
-0.04261230304837227,
-0.06253713369369507,
-0.04602045565843582,
-0.014643178321421146,
-0.046151965856552124,
-0.02462916634976864,
0.08082988113164902,
0.009626257233321667,
0.026069480925798416,
0.010368594899773598,
-0.06535317748785019,
-0.02928934432566166,
0.04206319525837898,
-0.045739591121673584,
0.1274554580450058,
0.02498514950275421,
-0.03461974859237671,
-0.049519024789333344,
0.03899183124303818,
0.06016109883785248,
-0.029946794733405113,
-0.026279479265213013,
-0.003394985804334283,
-0.029628565534949303,
-0.020893877372145653,
-0.05300517752766609,
0.03476966917514801,
-0.0022761726286262274,
0.1549016386270523,
0.061952948570251465,
0.009542109444737434,
0.024199552834033966,
-0.019403671845793724,
0.13486309349536896,
-0.035762086510658264,
0.006286219693720341,
-0.09331050515174866,
-0.08897274732589722,
-0.0753454864025116,
0.018997250124812126,
-0.0022288882173597813,
-0.0012506209313869476,
0.01443841215223074,
-0.05887199938297272,
-0.0023220337461680174,
-0.07296162098646164,
0.004180856514722109,
-0.03844606876373291,
0.0188441164791584,
0.07064119726419449,
0.03773785009980202,
0.06837617605924606,
-0.002086901105940342,
0.014347019605338573,
0.008755111135542393,
-0.006523404270410538,
0.060731250792741776,
0.04649055004119873,
-0.12542621791362762,
0.010400120168924332,
-0.07604780048131943,
-0.02420569211244583,
-0.04984256997704506,
0.04970616474747658,
-0.03156118094921112,
-0.01852516084909439,
0.04290217533707619,
0.04508763179183006,
0.025309011340141296,
-0.10232442617416382,
-0.08852594345808029,
0.008978494442999363,
0.08042874187231064,
0.07379166036844254,
-0.06318424642086029,
-0.07242636382579803,
0.011239180341362953,
0.0007071108557283878,
0.01722494699060917,
-0.10115691274404526,
0.068915955722332,
8.054969244518151e-32,
0.02949598617851734,
0.02950911968946457,
-0.02542518451809883,
-0.012200340628623962,
-0.05404828488826752,
0.009925905615091324,
0.02104354090988636,
-0.007382448762655258,
0.01980804093182087,
-0.11905606091022491,
-0.012302636168897152,
0.05866092070937157,
0.02372293919324875,
0.040925268083810806,
-0.092166468501091,
0.030598407611250877,
-0.011196515522897243,
0.05128234624862671,
-0.016152886673808098,
0.02656552381813526,
-0.054746363312006,
-0.009270324371755123,
-0.07134142518043518,
0.06562343239784241,
0.01789608970284462,
0.018860766664147377,
-0.046008311212062836,
0.050513312220573425,
-0.05466790497303009,
-0.026198599487543106,
0.0749790295958519,
0.05571383982896805,
-0.017284702509641647,
-0.027031907811760902,
0.04292077198624611,
0.0661478042602539,
0.04016312211751938,
-0.009623995050787926,
0.044787656515836716,
0.018306920304894447,
0.03075409308075905,
-0.0738309919834137,
0.0011666244827210903,
0.04060535132884979,
0.030407683923840523,
0.027164800092577934,
-0.0311365257948637,
-0.05029960721731186,
0.04827522486448288,
-0.0697794184088707,
-0.021082812920212746,
0.012328486889600754,
0.07846565544605255,
0.06130621209740639,
-0.015741880983114243,
-0.12175798416137695,
-0.0453181192278862,
-0.014929677359759808,
0.03182036057114601,
-0.028309429064393044,
-0.04845263063907623,
0.04890340194106102,
-0.008642401546239853,
-0.01327578630298376
] |
8
Chapter 1
Example 8
Factorise completely:
a x3 β 2x2 b x3 β 25x c x3 + 3x2 β 10xb x2 + 6 x + 8
= x2 + 2 x + 4 x + 8
= x(x
+ 2) + 4( x + 2)
= (x
+ 2)( x + 4)
c 6x2 β 11 x β 10
= 6x2 β 15 x + 4 x β 10
= 3x(2x
β 5) + 2(2 x β 5)
= (2 x
β 5)(3 x + 2)
d x2 β 25
= x2 β 52
= (x + 5)( x β 5)
e 4x2 β 9 y2
= 22x2 β 32y2
= (2 x + 3y)(2x β 3 y)
a x3 β 2x2 = x2(x β 2)
b x3 β 25 x = x(x2 β 25)
= x(x2 β 52)
= x(x + 5)( x β 5)
c x3 + 3 x2 β 10x = x (x2 + 3 x β 10)
= x(x + 5)( x β 2)
1 Factorise these expressions completely:a
4x + 8 b 6x β 24 c 20x + 15
d 2x2 + 4 e 4x2 + 20 f 6x2 β 18x
g x2 β 7x h 2x2 + 4x i 3x2 β x
j 6x2 β 2x k 10y2 β 5y l 35x2 β 28x
m x2 + 2x n 3y2 + 2y o 4x2 + 12x
p 5y2 β 20y q 9xy2 + 12x2y r 6abΒ β 2ab2
s 5x2 β 25xy t 12x2y ξ± 8xy2 u 15y β 20yz2
v 12x2 β 30 w xy2 β x2y x 12y2 β 4yxExercise 1Cx2 β 25 is the difference of two squares.This is the difference of two squares as the two
terms are x2 and 52.
The two x terms, 5x and β 5x, cancel each other out.acΒ =Β β60 and 4Β βΒ 15Β =Β β11Β =Β b.
Factorise.
This is the same as (2x)2Β βΒ (3y)2.
You canβt factorise this any further.
x is a common factor of x3 and β25x.
So take x outside the brackets.
Write the expression as a product of x and a
quadratic factor.
Factorise the quadratic to get three linear factors.acΒ =Β 8 and 2Β +Β 4Β =Β 6Β =Β b.
Factorise.
|
[
-0.03140981122851372,
0.08939728885889053,
0.0034476378932595253,
-0.05997445434331894,
0.03819393739104271,
0.07588260620832443,
-0.060193318873643875,
-0.009052691049873829,
-0.09045165777206421,
0.08356274664402008,
0.030681505799293518,
-0.07013620436191559,
0.07631919533014297,
-0.069104865193367,
-0.0038022769149392843,
-0.03740033134818077,
-0.019205572083592415,
0.04644966125488281,
-0.06252580881118774,
-0.03285639360547066,
0.03255636617541313,
-0.04009329900145531,
-0.00847314391285181,
-0.058702919632196426,
0.08788570016622543,
-0.05383944883942604,
0.015072684735059738,
-0.11629768460988998,
-0.025101246312260628,
-0.07638572156429291,
-0.009642326273024082,
0.07307953387498856,
0.08514254540205002,
0.002229229314252734,
0.041194502264261246,
0.040019724518060684,
0.004476419184356928,
0.0369059219956398,
-0.027432149276137352,
-0.030478231608867645,
-0.04760732129216194,
0.030093101784586906,
-0.016848638653755188,
0.03477846458554268,
0.03564753383398056,
-0.027928348630666733,
0.02789790742099285,
-0.008361178450286388,
0.04136122390627861,
-0.09493135660886765,
0.0008285745861940086,
-0.026022223755717278,
-0.003907144069671631,
0.03582170605659485,
-0.02677193470299244,
-0.006809568498283625,
-0.08429338037967682,
0.07712412625551224,
-0.017365816980600357,
-0.02311016619205475,
-0.043344397097826004,
-0.07243560254573822,
0.010396359488368034,
0.058049652725458145,
0.04343488812446594,
0.05800681933760643,
-0.04227032884955406,
-0.09410601109266281,
0.018014486879110336,
0.009447951801121235,
-0.04686904698610306,
0.02093929797410965,
-0.013807987794280052,
0.011880266480147839,
0.07569881528615952,
0.03692717105150223,
-0.02485332265496254,
-0.05980823189020157,
0.02590883895754814,
-0.0011445513227954507,
0.013132954016327858,
-0.01854502409696579,
0.08697525411844254,
0.006723157130181789,
0.061752889305353165,
-0.05503406003117561,
0.1217450350522995,
0.0009798690443858504,
0.04738498851656914,
-0.08096619695425034,
0.031084135174751282,
-0.012685705907642841,
0.016336921602487564,
-0.03998984768986702,
0.0026430857833474874,
-0.05594389885663986,
-0.013987281359732151,
-0.034680288285017014,
0.03134613484144211,
0.016742419451475143,
0.012469611130654812,
0.008516045287251472,
-0.01288633607327938,
-0.02027585171163082,
-0.04110689461231232,
-0.04884621873497963,
-0.03536393120884895,
-0.01506622601300478,
0.08044254779815674,
-0.09397616982460022,
-0.08670724183320999,
-0.04949043691158295,
-0.0017851664451882243,
-0.026631511747837067,
-0.013428916223347187,
-0.10267777740955353,
0.0909188836812973,
-0.009184692986309528,
0.04256594553589821,
0.019576840102672577,
0.015860499814152718,
-0.07720372080802917,
0.052707597613334656,
-0.03738059848546982,
0.002517768181860447,
-0.008544241078197956,
-0.007691010367125273,
-0.014929158613085747,
0.002992484252899885,
-0.12172231823205948,
0.03783677890896797,
0.004610596224665642,
-0.061155907809734344,
0.011228812858462334,
-0.04499822482466698,
-0.10807189345359802,
-0.06442823261022568,
0.07146201282739639,
-0.009741167537868023,
-0.010751740075647831,
0.01985897496342659,
0.09928740561008453,
0.039419203996658325,
0.05039764568209648,
-0.01671786606311798,
-0.04434295743703842,
-0.05461820960044861,
-0.044346705079078674,
-0.002105065854266286,
0.07676348090171814,
-0.0074170539155602455,
-0.025192243978381157,
-0.0392443910241127,
0.03292243927717209,
0.030582383275032043,
-0.06360423564910889,
0.019366765394806862,
-0.08538550138473511,
-0.020444409921765327,
-0.06848529726266861,
-0.12309188395738602,
0.01002291589975357,
-0.05600371211767197,
0.06739066541194916,
-0.009755231440067291,
-0.007226853631436825,
0.010517215356230736,
0.10705874860286713,
-0.008271642960608006,
-0.014034761115908623,
0.011256475932896137,
-0.03882560133934021,
0.031115349382162094,
0.09084294736385345,
0.008354807272553444,
-0.0210577342659235,
0.02213175594806671,
0.018085340037941933,
0.042841579765081406,
0.1177792102098465,
0.12140937149524689,
0.02956526353955269,
-0.03648926317691803,
0.05710005387663841,
-0.021635182201862335,
0.015151259489357471,
0.008787521161139011,
0.051429204642772675,
0.04155180603265762,
-0.05001227557659149,
0.010622248984873295,
-0.08103777468204498,
-0.05986136198043823,
0.03192644938826561,
-0.048543043434619904,
-0.024671342223882675,
0.0013545857509598136,
-0.09038769453763962,
-0.11907228827476501,
0.00023313265410251915,
0.028529731556773186,
0.026168253272771835,
0.08818238973617554,
-0.06304098665714264,
-0.04952669516205788,
0.0034888756927102804,
-0.05968021973967552,
-0.02750737965106964,
0.024672172963619232,
0.07650890201330185,
0.004434574861079454,
0.02276284247636795,
-0.014531434513628483,
-0.019606826826930046,
0.05254131928086281,
0.024127807468175888,
0.052688732743263245,
-0.1256566047668457,
-0.009265189059078693,
-0.04168526083230972,
0.042684156447649,
-0.09312335401773453,
0.004933900199830532,
-8.002431993876088e-33,
-0.09015711396932602,
-0.06308286637067795,
-0.051979854702949524,
-0.035096731036901474,
0.016051150858402252,
-0.08210530132055283,
0.003921248484402895,
-0.08938326686620712,
0.0943526104092598,
0.05609879642724991,
0.02310158871114254,
-0.032394856214523315,
-0.008312753401696682,
0.03652435541152954,
-0.04687011241912842,
-0.07076296955347061,
-0.05792385712265968,
0.04149257764220238,
0.05804196372628212,
-0.06616030633449554,
-0.01945253647863865,
0.009135209023952484,
0.011293778195977211,
0.00404586223885417,
0.026395371183753014,
-0.024690086022019386,
0.0300179123878479,
-0.11941302567720413,
0.09146042168140411,
0.04631577432155609,
-0.013343587517738342,
-0.0681021586060524,
0.03454911708831787,
0.0630001500248909,
-0.0227587278932333,
-0.06280140578746796,
-0.008339472115039825,
0.011209418997168541,
-0.046045251190662384,
-0.02425849810242653,
0.07616846263408661,
-0.00907166302204132,
0.01693623699247837,
0.02857886627316475,
-0.022025473415851593,
0.048634130507707596,
0.0011568287154659629,
0.03546834737062454,
-0.003764770459383726,
-0.007781469263136387,
-0.0037752867210656404,
-0.030476458370685577,
-0.026550838723778725,
0.011447289027273655,
-0.025905728340148926,
0.006620433181524277,
0.10595538467168808,
0.005232372786849737,
0.03839969262480736,
0.008303920738399029,
-0.07893417030572891,
0.002163176191970706,
0.014880509115755558,
0.03620227053761482,
-0.02216699719429016,
0.022960588335990906,
-0.027483204379677773,
0.02157135121524334,
-0.05356823280453682,
0.060728102922439575,
-0.03957553952932358,
0.11152105033397675,
-0.0673268586397171,
0.03983065485954285,
0.035611510276794434,
0.07573652267456055,
-0.006112849805504084,
0.04419563338160515,
-0.0639900341629982,
-0.04113086313009262,
-0.024792559444904327,
-0.04504403844475746,
-0.010662607848644257,
-0.09388028085231781,
-0.04675474017858505,
-0.009167949669063091,
0.15074144303798676,
0.10151725262403488,
0.017770199105143547,
-0.040947116911411285,
0.04126962646842003,
-0.06740907579660416,
0.10464519262313843,
-0.04629385471343994,
0.10666067153215408,
1.1033564959515903e-31,
0.006986203137785196,
-0.03692349046468735,
-0.10107725113630295,
0.03781908005475998,
0.009046792984008789,
-0.024989129975438118,
0.01900164969265461,
0.007357021793723106,
0.00456070713698864,
-0.07423152774572372,
0.06666120141744614,
0.060581307858228683,
-0.01064228918403387,
0.04340353235602379,
-0.04733264818787575,
-0.019660254940390587,
-0.06827406585216522,
-0.04382485896348953,
-0.0034320077393203974,
0.07599040865898132,
0.00665877852588892,
0.019419638440012932,
-0.0434289388358593,
-0.026063542813062668,
0.023077059537172318,
0.033163607120513916,
-0.023444348946213722,
0.034821927547454834,
0.07631365954875946,
0.04497547447681427,
0.04999912157654762,
-0.02778252214193344,
-0.07265716791152954,
-0.04191354289650917,
-0.012474055401980877,
0.022260405123233795,
0.11659444868564606,
0.10026192665100098,
-0.010205433703958988,
-0.012860450893640518,
-0.006502450909465551,
-0.05092482268810272,
-0.013455345295369625,
-0.03953107073903084,
-0.012430437840521336,
-0.04515639692544937,
-0.04040181264281273,
0.04607977345585823,
0.003962191753089428,
-0.043883711099624634,
-0.061917755752801895,
0.009356948547065258,
0.022802356630563736,
-0.015635550022125244,
0.012219159863889217,
-0.10035762935876846,
-0.028050635010004044,
-0.04596077278256416,
0.012968613766133785,
-0.09131879359483719,
0.005697743501514196,
0.07343334704637527,
-0.010082480497658253,
0.06186525523662567
] |
9Algebraic expressions
Write 4x4 β 13x2 + 9 as the product of four linear factors.Challenge2 Factorise:
a x2 + 4x b 2x2 + 6x c x2 + 11x + 24
d x2 + 8x + 12 e x2 + 3xΒ β 40 f x2 β 8x + 12
g x2 + 5x + 6 h x2 β 2xΒ β 24 i x2 β 3xΒ β 10
j x2 +Β xΒ β 20 k 2x2 + 5xΒ + 2 l 3x2 + 10x β 8
m 5x2 β 16xΒ + 3 n 6x2 β 8x β 8
o 2x2 + 7xΒ β 15 p 2x4 + 14x2 + 24
q x2 β 4 r x2 β 49
s 4x2 β 25 t 9x2 β 25y2 u 36x2 β 4
v 2x2 β 50 w 6x2 β 10xΒ + 4 x 15x2 + 42xΒ β 9
3 Factorise completely:a
x3 + 2x b x3 β x2 + x c x3 β 5x
d x3 β 9x e x3 β x2 β 12x f x3 + 11x2 + 30x
g x3 β 7x2 + 6x h x3 β 64x i 2x3 β 5x2 β 3x
j 2x3 + 13x2 + 15x k x3 β 4x l 3x3 + 27x2 + 60x
4 Factorise completel
y x4 β y4. (2 marks)
5 Factorise completel
y 6x3 + 7x2 β 5x. (2 marks) For part n , ta ke 2 out as a common
factor first. For part p , let yΒ =Β x2.Hint
Watch out for terms that can be written as a
function of a function: x4 = (x2)2Problem-solving P
E
1.4 Negative and fractional indic es
Indices can be negative numbers or fractions.
x 1 _ 2 Γ x 1 _ 2 = x 1 _ 2 + 1 _ 2 = x 1 = x,
similarly x 1 __ n Γ x 1 __ n Γ . . . Γ x 1 __ n = x 1 __ n + 1 __ n +...+ 1 __ n = x 1 = x
n terms
β You can use the laws of indices with any rational power.
β’ a 1 __ m = m ββ―__
a
β’ a n __ m = m ββ―___ a n
β’ a βm = 1 ___ a m
β’ a 0 = 1 β«
βͺβͺβ¬βͺβͺβ Ratio nal
numbers are those that
can be written as a __ b where
a and b are integers.Notation
a 1 _ 2 = ββ―__
a is the
positive square root of a .
For example 9 1 _ 2 = ββ―__
9 = 3
but
9 1 _ 2 β β3 .Notation
|
[
-0.006527918856590986,
0.12704862654209137,
-0.013633491471409798,
-0.02474166266620159,
-0.008003801107406616,
-0.05678652226924896,
-0.045383717864751816,
-0.0665820762515068,
-0.05042552575469017,
0.031354621052742004,
-0.07400527596473694,
-0.11872166395187378,
-0.039546217769384384,
0.020131956785917282,
-0.007070524618029594,
-0.04442013055086136,
-0.03932756558060646,
0.07595225423574448,
-0.09360068291425705,
0.003914757631719112,
0.08475098758935928,
-0.11320021748542786,
0.010731063783168793,
-0.012338262051343918,
0.058468956500291824,
-0.029294686391949654,
-0.011946451850235462,
-0.0697731301188469,
0.01851293072104454,
-0.01146312989294529,
0.030559899285435677,
0.046905964612960815,
0.1329599916934967,
0.0002114483795594424,
0.03643547743558884,
0.043884288519620895,
0.016552206128835678,
0.05232476815581322,
-0.020131541416049004,
-0.02079305425286293,
-0.034229155629873276,
-0.01594497635960579,
-0.032140087336301804,
-0.018797757104039192,
0.03156489133834839,
-0.12531419098377228,
0.06693769991397858,
0.06158200278878212,
0.06746575236320496,
-0.07204876095056534,
0.07097727060317993,
-0.0413391999900341,
-0.04256747290492058,
0.07506176084280014,
0.0374910943210125,
-0.04873093217611313,
-0.008783487603068352,
0.09831272065639496,
0.05967756360769272,
-0.04462035745382309,
-0.028934575617313385,
-0.07753307372331619,
-0.01091474387794733,
0.054715596139431,
-0.038486432284116745,
0.01422900427132845,
-0.09720258414745331,
-0.019235437735915184,
0.007749724201858044,
-0.01824495568871498,
-0.003005172125995159,
0.006200671661645174,
-0.00006972491246415302,
-0.04596159979701042,
0.035276930779218674,
0.038946669548749924,
-0.010454528033733368,
-0.04776397719979286,
0.04281863570213318,
-0.044652059674263,
0.021934743970632553,
0.0020136279053986073,
0.06080535426735878,
-0.01547784823924303,
0.0287020206451416,
0.01808498054742813,
0.09551675617694855,
0.05435661971569061,
0.05748091638088226,
0.002915628720074892,
-0.023437565192580223,
-0.02031528949737549,
0.011927508749067783,
-0.03534378483891487,
-0.04893272742629051,
-0.07354176789522171,
-0.036219775676727295,
-0.03621796891093254,
-0.06271526962518692,
0.08279544860124588,
0.0385882742702961,
-0.01858672685921192,
-0.06366139650344849,
-0.04481960088014603,
-0.03163529187440872,
-0.011501770466566086,
0.021985309198498726,
-0.013548403978347778,
0.0370500348508358,
-0.05158839002251625,
-0.057525377720594406,
0.027686424553394318,
0.004830023739486933,
-0.03283967450261116,
0.00872599333524704,
-0.03897137567400932,
0.07216007262468338,
-0.0192671287804842,
0.011454911902546883,
-0.048104628920555115,
-0.015571480616927147,
-0.025438375771045685,
0.07964705675840378,
-0.029482131823897362,
0.06685914099216461,
0.015163969248533249,
-0.07024502754211426,
-0.015059402212500572,
-0.0026080862153321505,
-0.06892266124486923,
0.03214818611741066,
0.011348441243171692,
0.02137463353574276,
0.005531011614948511,
-0.03589605167508125,
-0.06125475838780403,
-0.038968734443187714,
0.10749971121549606,
-0.04371437430381775,
-0.034977346658706665,
0.04711395874619484,
0.03737382963299751,
0.0287599116563797,
0.04711134359240532,
-0.019540442153811455,
-0.04588296264410019,
0.01913699507713318,
-0.10246351361274719,
-0.017883995547890663,
0.08420722931623459,
0.06388580799102783,
-0.024444200098514557,
-0.10810832679271698,
0.024459583684802055,
0.06322776526212692,
-0.07609810680150986,
-0.03425828367471695,
-0.10703015327453613,
-0.04419281706213951,
-0.07323624938726425,
-0.010696524754166603,
-0.016210371628403664,
0.06234917417168617,
0.07381066679954529,
0.02639317698776722,
0.04798446595668793,
-0.012913601472973824,
0.03899557515978813,
-0.006331511773169041,
0.01133958250284195,
0.030374079942703247,
0.010874217376112938,
0.013667099177837372,
0.0021586776711046696,
0.06324626505374908,
-0.04162083938717842,
0.01169859804213047,
-0.04547417163848877,
-0.037359848618507385,
0.13456162810325623,
0.07883787155151367,
-0.00819212943315506,
-0.04827379807829857,
-0.034908752888441086,
0.02631464973092079,
0.020354527980089188,
0.07269477099180222,
0.04226084053516388,
-0.10076580941677094,
-0.12057989835739136,
0.06583882868289948,
-0.023012589663267136,
-0.0788324624300003,
-0.03730708733201027,
-0.09190905839204788,
0.03410732001066208,
-0.08200787007808685,
-0.019714314490556717,
-0.06643948704004288,
-0.01233238261193037,
0.06032600998878479,
0.0008653350523672998,
0.05901511013507843,
-0.07395944744348526,
0.009893648326396942,
0.013408444821834564,
0.007537728175520897,
-0.021794212982058525,
0.05424579605460167,
0.08021927624940872,
-0.040929924696683884,
0.02691720612347126,
0.01689060591161251,
0.04604863375425339,
0.008575948886573315,
-0.044245921075344086,
0.10278625786304474,
-0.021810341626405716,
0.011131657287478447,
-0.0832650363445282,
-0.02088761515915394,
-0.08856870979070663,
0.02686965838074684,
-1.354548882949003e-32,
-0.054098960012197495,
-0.0528164766728878,
-0.02397812157869339,
-0.030213113874197006,
0.019351376220583916,
-0.017960546538233757,
0.0318109355866909,
-0.0637444332242012,
0.0982903316617012,
0.036426249891519547,
0.06485763937234879,
-0.035804931074380875,
0.02411007694900036,
-0.002027041045948863,
-0.04781891033053398,
-0.07688254117965698,
-0.0607263445854187,
0.03958626091480255,
0.07106518745422363,
-0.07787766307592392,
0.013670592568814754,
0.06311895698308945,
0.0007806305657140911,
0.044760752469301224,
-0.027267983183264732,
-0.008151072077453136,
0.023512747138738632,
-0.06094813346862793,
-0.009926620870828629,
0.018583722412586212,
0.040183451026678085,
-0.04312673583626747,
0.0586247593164444,
0.05745500698685646,
0.05744878947734833,
-0.028464170172810555,
-0.01885894313454628,
0.016929278150200844,
0.018489575013518333,
-0.02633710391819477,
0.056984491646289825,
0.05392391234636307,
0.02507646009325981,
0.026361072435975075,
-0.03630346432328224,
-0.0035707885399460793,
0.006023033522069454,
0.05839734151959419,
0.016293048858642578,
-0.002191005740314722,
-0.059942182153463364,
-0.07874374836683273,
-0.08215410262346268,
-0.046935781836509705,
-0.020311415195465088,
0.006619047839194536,
0.14418178796768188,
-0.07059299945831299,
0.0381837822496891,
0.04100234434008598,
-0.10932512581348419,
-0.05600934103131294,
0.03354484215378761,
0.0038417258765548468,
-0.007833532057702541,
0.026410991325974464,
0.010230590589344501,
0.021647891029715538,
0.015232747420668602,
-0.01345091499388218,
-0.043377529829740524,
0.12292373180389404,
-0.14828917384147644,
0.057873956859111786,
0.018038427457213402,
-0.07599892467260361,
-0.009360010735690594,
0.0837668925523758,
-0.07045667618513107,
-0.0645173117518425,
-0.028892992064356804,
-0.030961448326706886,
0.03412814438343048,
0.003754280973225832,
-0.03516082093119621,
0.015966925770044327,
0.07795488834381104,
0.08016270399093628,
-0.030233239755034447,
-0.038474977016448975,
0.07825227081775665,
0.05602341145277023,
0.009398295544087887,
-0.044367626309394836,
0.06269887834787369,
1.0500130325090152e-31,
0.02191140688955784,
0.05802540481090546,
-0.115666963160038,
-0.05589957907795906,
-0.00614233547821641,
-0.01808236353099346,
0.055843424052000046,
-0.05006200075149536,
0.02268298715353012,
-0.10082840174436569,
0.058237407356500626,
0.029444413259625435,
-0.048828110098838806,
0.014952464960515499,
-0.0669650062918663,
-0.02145576849579811,
-0.04545110836625099,
0.016885900869965553,
-0.014742407947778702,
0.01711258850991726,
-0.0223811287432909,
0.012986182235181332,
-0.034922633320093155,
-0.010314174927771091,
0.03180934488773346,
0.06069374829530716,
-0.012387275695800781,
0.01904057152569294,
-0.05165398120880127,
-0.04529190808534622,
0.07469383627176285,
-0.045095644891262054,
-0.026447653770446777,
-0.03646247833967209,
-0.011560363695025444,
0.05835140123963356,
0.06765963137149811,
0.0701414942741394,
0.013281297869980335,
0.044313304126262665,
0.06826300173997879,
-0.06199970841407776,
0.014118473045527935,
-0.024861667305231094,
0.02412319742143154,
-0.039682503789663315,
-0.018751224502921104,
-0.03637474775314331,
0.04459432139992714,
-0.023606665432453156,
-0.06886538863182068,
-0.06923600286245346,
-0.008430616930127144,
-0.020960571244359016,
-0.038612402975559235,
-0.022523317486047745,
0.041290998458862305,
0.027782388031482697,
-0.08138466626405716,
-0.09013080596923828,
-0.01335575059056282,
0.031469348818063736,
-0.04981575906276703,
-0.04303491860628128
] |
10
Chapter 1
Example 9
Simplify:
a x 3 ___ x β3 b x1
2 Γ x32
c (x3)23
d 2x1.5Β Γ·Β 4xβ0.25 e 3 ββ―______ 125 x 6 f 2 x 2 β x _______ x 5
a x 3 ____ x β3 = x3 β (β3) = x6
b x1
2 Γ x3
2 = x1
2 ξ±Β 32 = x2
c (x3)23 =Β x3 ξ³Β 23 =Β x2
d 2x1.5Β ξ΄Β 4 xβ0.25 = 1 __ 2 x1.5Β β (β0 .25) = 1 __ 2 x1.75
e 3 β _____ 125 x 6 = ( 125 x6 ) 1 __ 3
= (12
5 ) 1 __ 3 (x6 ) 1 __ 3 = 3 ββ―_____ 125 ( x 6 Γ 1 __ 3 ) = 5 x2
f 2 x 2 β x ______ x 5 = 2 x 2 ____ x 5 β x ___ x 5
= 2 Γ
x2 β 5 β x1 β 5 = 2xβ3 β xβ4
= 2 ___ x 3 β 1 ___ x 4 Use the rule amΒ Γ·Β anΒ =Β am β n.
Evaluate:
a 9 1 _ 2 b 6 4 1 _ 3 c 4 9 3 _ 2 d 2 5 β 3 _ 2 Example 10
a 9 1 __ 2 = ββ―__
9 = 3
b 6 4 1 __ 3 = 3 ββ―___ 64 = 4
c 4 9 3 __ 2 = ( ββ―___ 49 ) 3
73 = 343
d 2 5 β 3 __ 2 = 1 ____
2 5 3 __ 2 = 1 ______
( ββ―___ 25 ) 3
= 1 ___ 53 = 1 _____ 125 Using a 1 __ m = m ββ―__
a . 9 1 _ 2 = ββ―__
9
Using a n __ m = m ββ―__ an .
This means the square root o
f 49, cubed.
Using aβm = 1 ___ am This could also be written as ββ―__
x .
Use the rule amΒ ΓΒ anΒ =Β am + n.
Use the rule (am)nΒ =Β amn.
Use the rule amΒ Γ·Β anΒ =Β am β n.
1.5Β βΒ (β0.25)Β =Β 1.75
Using a 1 __ m = m ββ―__
a .
Divide each term of the numerator by x5.
Using a βm = 1 ___ a m
This means the cube root of 64.
Use your calculator to enter
ne
gative and fractional powers.Online
|
[
0.0012006928445771337,
0.07275675982236862,
0.02219429239630699,
-0.07081831991672516,
-0.007893268950283527,
0.04595699906349182,
0.01653374917805195,
-0.025362949818372726,
-0.07866908609867096,
0.0866442546248436,
-0.04205698147416115,
-0.029072867706418037,
0.05228177085518837,
-0.07409974187612534,
-0.03485164791345596,
-0.026161598041653633,
-0.08974213153123856,
0.12637841701507568,
-0.13320548832416534,
-0.007612238638103008,
0.024505827575922012,
-0.07527699321508408,
-0.016726849600672722,
-0.016942955553531647,
0.11248107999563217,
0.0029743395280092955,
0.02169600874185562,
-0.054376520216464996,
-0.07032939046621323,
-0.139139786362648,
0.011713131330907345,
0.06067109853029251,
0.11652515828609467,
-0.058248121291399,
0.005861049052327871,
0.005758277140557766,
-0.005431385710835457,
0.03790279105305672,
-0.031582433730363846,
0.019555646926164627,
-0.04120176285505295,
0.08312835544347763,
0.0506097786128521,
0.03436220437288284,
0.03511589765548706,
-0.026522260159254074,
-0.03324465453624725,
-0.020593443885445595,
-0.01154673844575882,
-0.06628967076539993,
0.03702152147889137,
-0.031487852334976196,
-0.07717609405517578,
-0.02972009778022766,
-0.04103393480181694,
-0.12193376570940018,
-0.0869356170296669,
0.04176078736782074,
-0.04554339870810509,
-0.04310076683759689,
-0.019668247550725937,
-0.0401654914021492,
0.0522836297750473,
0.04730615392327309,
-0.03575724735856056,
0.04369312897324562,
0.0005765332607552409,
-0.06217721849679947,
-0.0217667818069458,
0.09900877624750137,
-0.102043017745018,
0.03148364648222923,
-0.06402590870857239,
0.00008762506331549957,
0.08803551644086838,
-0.004214219283312559,
-0.05244268476963043,
-0.01946542225778103,
-0.036425720900297165,
-0.0026760140899568796,
-0.04728659614920616,
0.036610059440135956,
0.10661182552576065,
-0.011433962732553482,
0.06256472319364548,
-0.04860007390379906,
0.09562370181083679,
-0.029147831723093987,
0.041524115949869156,
-0.033245500177145004,
0.016196317970752716,
-0.03993632644414902,
-0.04610380157828331,
-0.024172907695174217,
-0.0819520503282547,
0.004987175110727549,
-0.0016868450911715627,
-0.06904897838830948,
0.04274281859397888,
0.044342681765556335,
0.0527496263384819,
0.03863057494163513,
0.013474658131599426,
0.028802871704101562,
-0.048686519265174866,
-0.08578450232744217,
0.030763857066631317,
-0.009473907761275768,
0.11821705847978592,
-0.10729129612445831,
-0.029078561812639236,
-0.01399755384773016,
-0.032343946397304535,
-0.05189889296889305,
-0.001075825421139598,
-0.08329596370458603,
0.011361450888216496,
0.018126746639609337,
0.040323566645383835,
-0.006720100529491901,
0.023733103647828102,
-0.03253784403204918,
0.041541893035173416,
0.02231699973344803,
0.01061415858566761,
0.002589087700471282,
-0.018626809120178223,
0.04209041967988014,
-0.06887537986040115,
-0.06185976043343544,
-0.050295643508434296,
-0.02900722809135914,
-0.03874651715159416,
0.03622502088546753,
-0.049846239387989044,
-0.02043762244284153,
-0.053620751947164536,
0.09577804803848267,
-0.033222686499357224,
0.019055107608437538,
0.043722644448280334,
0.07139352709054947,
0.03952978923916817,
0.028005916625261307,
0.0096384696662426,
-0.008611614815890789,
-0.03674504905939102,
-0.03751296550035477,
0.016640130430459976,
0.07505758851766586,
0.026270873844623566,
0.0030888060573488474,
0.04381079226732254,
0.06323804706335068,
-0.0513828881084919,
-0.04156581684947014,
0.015391809865832329,
-0.054155655205249786,
-0.005696367938071489,
-0.04478565603494644,
-0.0841493308544159,
0.04616471752524376,
-0.04625852406024933,
0.057123564183712006,
0.03119475580751896,
0.02171468734741211,
0.031128615140914917,
0.030364129692316055,
-0.05789773538708687,
0.009207898750901222,
0.04127943515777588,
-0.0602261908352375,
0.07029636949300766,
0.08754205703735352,
-0.012626410461962223,
0.017545461654663086,
0.07167884707450867,
0.017297321930527687,
0.010942679829895496,
0.0696374848484993,
0.11687424778938293,
-0.019336670637130737,
0.015268571674823761,
0.034707989543676376,
-0.036268118768930435,
-0.03565150126814842,
-0.04797732084989548,
0.02694918029010296,
0.013088972307741642,
-0.03184123709797859,
-0.004096062853932381,
-0.04191393032670021,
-0.021379897370934486,
0.025208113715052605,
-0.07691662758588791,
-0.04404136538505554,
-0.0006956458673812449,
-0.08404494822025299,
-0.11376355588436127,
0.06951115280389786,
0.019827518612146378,
-0.04392700269818306,
0.10930338501930237,
-0.02978786639869213,
-0.06938944011926651,
0.013761842623353004,
-0.03460680693387985,
-0.06945377588272095,
0.011968886479735374,
0.05575226619839668,
-0.0856102779507637,
-0.035165347158908844,
-0.07562961429357529,
-0.02995847351849079,
0.012146375142037868,
-0.0022369767539203167,
0.04623992368578911,
-0.04457050561904907,
-0.0532396137714386,
-0.013870329596102238,
-0.028497379273176193,
-0.03405286371707916,
0.025995615869760513,
-1.2087844243059716e-32,
-0.03469226136803627,
0.04684053733944893,
-0.10916589945554733,
-0.02096669375896454,
0.003878586459904909,
-0.039613693952560425,
0.07762094587087631,
-0.09491312503814697,
0.02204868011176586,
0.036019373685121536,
-0.02409622259438038,
0.0011037682415917516,
-0.03791190683841705,
0.0066130454652011395,
-0.07451088726520538,
-0.02911374904215336,
-0.041521359235048294,
-0.015249514020979404,
0.007496409118175507,
-0.051807183772325516,
-0.008123323321342468,
0.03446206450462341,
-0.021661721169948578,
0.026553519070148468,
-0.026606224477291107,
0.05604252219200134,
-0.030630553141236305,
-0.051655497401952744,
0.08429162949323654,
0.05226491764187813,
-0.006822188384830952,
-0.01242610439658165,
0.05695018544793129,
0.06518331170082092,
0.00026512020849622786,
-0.10070089250802994,
0.015781046822667122,
0.03830671310424805,
-0.013724486343562603,
0.0005272459238767624,
0.08860629796981812,
0.019636215642094612,
0.019219020381569862,
0.0019183047115802765,
0.023013006895780563,
0.06927046179771423,
0.04958009719848633,
0.06322037428617477,
0.027143018320202827,
0.006154679227620363,
-0.032819539308547974,
-0.044383712112903595,
-0.01477737445384264,
0.007272864691913128,
0.0027120693121105433,
0.03337367996573448,
0.054290395230054855,
-0.0687355026602745,
0.04576524719595909,
0.012323564849793911,
-0.08541836589574814,
0.014942005276679993,
0.07278572022914886,
0.08018837869167328,
-0.015371754765510559,
0.06647342443466187,
-0.06602901965379715,
-0.008808870799839497,
-0.04491812735795975,
-0.0033945359755307436,
-0.04591847583651543,
0.1272500604391098,
-0.04956073313951492,
-0.08909180760383606,
0.0105745829641819,
0.014484732411801815,
0.001077020075172186,
-0.002829080680385232,
-0.028892887756228447,
-0.03094671666622162,
-0.0810788944363594,
-0.04272180050611496,
-0.010091383941471577,
-0.04381713643670082,
-0.04890889301896095,
-0.016828490421175957,
0.09264376759529114,
0.11749549955129623,
0.049340661615133286,
0.03139989450573921,
0.011209576390683651,
0.011049891822040081,
0.09052015095949173,
-0.02344883792102337,
0.046201881021261215,
1.1190422925689629e-31,
0.016723934561014175,
-0.033822689205408096,
-0.06955695152282715,
-0.011104810051620007,
-0.014616720378398895,
0.01694607548415661,
0.017829688265919685,
-0.0036835831124335527,
0.044142767786979675,
-0.04038256034255028,
0.029307633638381958,
0.04593834653496742,
-0.03567313402891159,
0.05520166456699371,
-0.09898854047060013,
-0.044973328709602356,
-0.07007598876953125,
-0.022464832291007042,
0.018839584663510323,
0.06834810227155685,
-0.012360535562038422,
0.023303020745515823,
-0.09093405306339264,
0.01882041059434414,
0.05717790499329567,
0.05996774882078171,
-0.09988594055175781,
0.03146275877952576,
0.03018992394208908,
-0.013394241221249104,
0.04104786366224289,
-0.012037476524710655,
0.03513951227068901,
-0.005798024591058493,
0.019716016948223114,
0.05648484081029892,
0.1364002823829651,
0.04474317282438278,
-0.021216031163930893,
0.0009137506713159382,
0.015223152004182339,
-0.06431837379932404,
-0.05582674965262413,
-0.02200646698474884,
0.03510845825076103,
-0.10525346547365189,
-0.08362342417240143,
-0.0274447463452816,
0.03381701931357384,
-0.05532168969511986,
-0.031162409111857414,
-0.004987399093806744,
-0.026234440505504608,
-0.01284478884190321,
-0.011242455802857876,
-0.046373747289180756,
-0.024614382535219193,
-0.00010007434320868924,
-0.00717816362157464,
-0.027167506515979767,
-0.04191429167985916,
0.10603191703557968,
-0.07451647520065308,
0.07569515705108643
] |
11Algebraic expressions
1 Simplify:
a x3 Γ· xβ2 b x5 Γ· x7 c x 3 _ 2 Γ x 5 _ 2
d (x2 ) 3 _ 2 e (x3 ) 5 _ 3 f 3x0.5 Γ 4xβ0.5
g 9 x 2 _ 3 Γ· 3 x 1 _ 6 h 5 x 7 _ 5 Γ· x 2 _ 5 i 3x4 Γ 2xβ5
j ββ―__
x Γ 3 ββ―__
x k ( ββ―__
x )3 Γ ( 3 ββ―__
x )4 l ( 3 ββ―__
x )2 _____ ββ―__
x
2 Eva
luate:
a 2 5 1 _ 2 b 8 1 3 _ 2 c 2 7 1 _ 3
d 4β2 e 9 β 1 _ 2 f (β 5)β3
g ( 3 _ 4 ) 0 h 129 6 3 _ 4 i ( 25 __ 16 ) 3
2
j ( 27 __ 8 ) 2
3 k ( 6 _ 5 ) β1 l ( 343 ___ 512 ) β2
3
3 Simplify:
a (64x10)1
2 b 5 x 3 β 2 x 2 ________ x 5 c (125x12)1
3 d x + 4 x 3 _______ x 3
e 2x + x 2 _______ x 4 f ( 4 __ 9 x4) 3
2 g 9 x 2 β 15 x 5 _________ 3 x 3 h 5x + 3 x 2 ________ 15 x 3
4 a Find the value of
8 1 1 _ 4 . (1 mark)
b Simplify x(2 x β 1 _ 3 )4. (2 marks)
5 Given tha
t y = 1 __ 8 x 3 express each of the following in the form k x n , where k and n are constants.
a y 1
3 (2 marks)
b 1 __ 2 y β2 (2 marks)E
EExercise 1Da y 1 _ 2 = ( 1 __ 16 x 2 ) 1 _ 2
= 1 ___ ββ―___ 16 x 2 Γ 1 _ 2 = x __ 4
b 4yβ1 = 4 ( 1 __ 16 x 2 ) β1
= 4 ( 1 __ 16 ) β1
x 2 Γ (β1) = 4 Γ 16 xβ2
= 64 xβ2Substitute y = 1 ___ 16 x 2 into y 1 _ 2 .
( 1 ___ 16 ) 1 _ 2
= 1 ____
β ___ 16 and ( x 2 ) 1 _ 2 = x 2 Γ 1 _ 2 Given that y = 1 __ 16 x2 express each of the following in the form k x n , where k and n are constants.
a y 1 __ 2 b 4 y β1 Example 11
Check that your answers are in the correct form.
If k and n are constants they could be positive or negative, and they could be integers, fractions or surds.Problem-solving ( 1 ___ 16 ) β1
= 16 and x2 Γ β1 = xβ2
|
[
-0.03872751444578171,
0.08569058030843735,
0.06746622174978256,
-0.05809372290968895,
-0.056650757789611816,
-0.04730015620589256,
0.06453460454940796,
-0.0783417597413063,
-0.09844944626092911,
0.040205925703048706,
-0.02584327943623066,
-0.07831025868654251,
-0.017865903675556183,
-0.00847869087010622,
0.03832695260643959,
0.011568891815841198,
-0.02539723925292492,
0.06765130907297134,
-0.16374531388282776,
-0.021909521892666817,
0.09167353808879852,
-0.09490151703357697,
0.007592870853841305,
-0.02278941124677658,
-0.007156990002840757,
-0.025895707309246063,
0.01796688698232174,
0.004686517640948296,
-0.04485367611050606,
-0.028797943145036697,
-0.004735749680548906,
0.0075421761721372604,
0.14838483929634094,
-0.10246611386537552,
0.023404739797115326,
-0.005796391982585192,
-0.01191709190607071,
0.04470417648553848,
-0.005733496509492397,
-0.013832001946866512,
0.024900875985622406,
0.04799412935972214,
0.05617911368608475,
-0.003330420469865203,
0.007711965590715408,
-0.050305042415857315,
0.008687124587595463,
-0.05109139904379845,
-0.05105152353644371,
-0.0314004048705101,
0.04454849660396576,
-0.02099466696381569,
-0.11814701557159424,
0.016981685534119606,
0.023271877318620682,
-0.1447940170764923,
0.008095710538327694,
0.017277449369430542,
-0.005868660286068916,
-0.009811618365347385,
-0.03674229234457016,
-0.042505111545324326,
0.01662781462073326,
0.05221094936132431,
-0.07764716446399689,
0.09846825897693634,
0.057207390666007996,
0.04518701136112213,
0.008600864559412003,
0.030102983117103577,
0.03273436054587364,
-0.001793519128113985,
-0.018373344093561172,
-0.05548254773020744,
0.09115418791770935,
0.023192163556814194,
-0.07421958446502686,
-0.11587177962064743,
-0.011361122131347656,
-0.04713805392384529,
-0.04086698591709137,
-0.014084727503359318,
0.06108077988028526,
-0.047283876687288284,
0.005835606716573238,
0.035714324563741684,
0.05674551799893379,
0.06614769250154495,
0.017701007425785065,
-0.038945235311985016,
0.0795418992638588,
-0.04658303037285805,
-0.016350867226719856,
-0.07323656231164932,
-0.07683669030666351,
-0.058355193585157394,
0.012247346341609955,
-0.08964106440544128,
-0.03500017151236534,
0.08438917994499207,
0.012082023546099663,
-0.0061949179507792,
0.012165805324912071,
0.008578039705753326,
-0.01799747720360756,
-0.08614316582679749,
-0.028821110725402832,
0.007259284611791372,
0.09611381590366364,
-0.02820182405412197,
0.008065207861363888,
-0.05450306460261345,
0.012506314553320408,
-0.009961780160665512,
0.003801670391112566,
0.033582042902708054,
0.06715232133865356,
-0.02846069447696209,
0.06911218166351318,
-0.038972511887550354,
-0.01673870161175728,
-0.04774308577179909,
0.05027972161769867,
0.0053743827156722546,
0.043326642364263535,
-0.04433736205101013,
-0.0426371693611145,
0.03684676066040993,
-0.049384381622076035,
0.010993153788149357,
-0.031890977174043655,
0.032243482768535614,
-0.059572603553533554,
0.006430283188819885,
-0.05425889790058136,
-0.10261014848947525,
-0.022213581949472427,
0.049395717680454254,
-0.0462360680103302,
0.059725381433963776,
0.013440800830721855,
0.044097986072301865,
0.02177506685256958,
0.06435202062129974,
-0.017243381589651108,
0.005749899428337812,
-0.013154065236449242,
-0.024983065202832222,
-0.004827468656003475,
0.05764976143836975,
0.03108925372362137,
0.007758956402540207,
-0.06340325623750687,
0.026555392891168594,
-0.008058606646955013,
-0.044695526361465454,
-0.010745050385594368,
-0.0807575136423111,
-0.0704505518078804,
-0.007028772961348295,
0.04423359036445618,
-0.02772647701203823,
0.005392289720475674,
0.07595977187156677,
0.04515622928738594,
0.02818029560148716,
0.052087631076574326,
0.002327041933313012,
-0.03948267921805382,
0.038767702877521515,
0.013689090497791767,
0.03718894347548485,
0.02603868767619133,
-0.008361898362636566,
-0.00018274938338436186,
-0.038838569074869156,
0.13815709948539734,
-0.05089591071009636,
0.0076580531895160675,
0.014522726647555828,
0.15008240938186646,
-0.07627961784601212,
0.014889507554471493,
0.03492657467722893,
-0.09003560990095139,
-0.0022886500228196383,
-0.030214082449674606,
0.014265049248933792,
-0.018635127693414688,
-0.07494256645441055,
0.05093107372522354,
-0.06262947618961334,
-0.09830738604068756,
-0.024182766675949097,
-0.04416946694254875,
-0.03502744063735008,
-0.13226498663425446,
-0.07545042037963867,
-0.09928502887487411,
0.06755902618169785,
0.0753215104341507,
-0.004094465170055628,
0.0831776037812233,
-0.018736643716692924,
0.012023323215544224,
-0.01056150533258915,
-0.009877554140985012,
-0.05354329198598862,
0.030715543776750565,
0.0667136088013649,
-0.07260686904191971,
0.015122114680707455,
0.0012901200680062175,
0.03194446489214897,
0.03853162005543709,
-0.03261873126029968,
-0.015348454937338829,
0.03898055478930473,
-0.043333277106285095,
-0.026667162775993347,
-0.005682564340531826,
-0.04236431419849396,
0.08029494434595108,
-1.9183387488064194e-32,
0.0005528049077838659,
0.03237137943506241,
-0.08763888478279114,
-0.029294807463884354,
-0.00433829240500927,
-0.01967194676399231,
0.04659745842218399,
-0.08303290605545044,
0.01690683327615261,
0.007210809271782637,
0.01294528879225254,
0.02191229909658432,
0.025425394997000694,
-0.0030407619196921587,
-0.06211903318762779,
-0.04263416305184364,
0.015922926366329193,
-0.020412519574165344,
-0.02965387888252735,
-0.05408795177936554,
-0.00876035075634718,
0.04983134940266609,
-0.034535836428403854,
0.053717050701379776,
-0.08316440135240555,
0.047615259885787964,
0.015529310330748558,
-0.03993261232972145,
0.040815845131874084,
0.0367092601954937,
0.04533207044005394,
-0.03563600778579712,
0.0634225606918335,
0.04547790437936783,
0.02276543155312538,
-0.07144745439291,
0.040437713265419006,
0.008381253108382225,
0.017322426661849022,
-0.023375723510980606,
0.022634057328104973,
-0.01818886585533619,
0.08949851244688034,
-0.03882579877972603,
0.036049939692020416,
-0.013100815936923027,
-0.020955484360456467,
0.10563284158706665,
0.005244196858257055,
-0.046394284814596176,
-0.037504374980926514,
-0.09688424319028854,
-0.1363639086484909,
-0.040406838059425354,
-0.031776949763298035,
0.030331246554851532,
0.04030793905258179,
-0.1138482540845871,
0.0841364860534668,
0.021453334018588066,
-0.0515882708132267,
-0.05103525519371033,
0.04315751791000366,
0.02647005394101143,
-0.03903915360569954,
0.058945927768945694,
-0.027926085516810417,
-0.0009218469494953752,
-0.0435967855155468,
-0.0015471752267330885,
0.031000513583421707,
0.08278009295463562,
-0.14311903715133667,
-0.05374949052929878,
-0.0102357417345047,
0.009690138511359692,
-0.011425658129155636,
0.05387221276760101,
-0.013878661207854748,
-0.067752905189991,
-0.03145090118050575,
-0.0035617200192064047,
0.025586498901247978,
-0.019920097663998604,
-0.08870088309049606,
-0.027426298707723618,
0.04525216668844223,
0.03549191728234291,
-0.02662612684071064,
-0.029086248949170113,
0.08850478380918503,
0.07935048639774323,
0.07662107050418854,
-0.0415080264210701,
0.05817725881934166,
1.2806883998818934e-31,
-0.013051903806626797,
0.017433881759643555,
-0.026390502229332924,
-0.04063039273023605,
0.040501661598682404,
-0.032850269228219986,
-0.0017490130849182606,
-0.061512451618909836,
0.04940035566687584,
-0.058998752385377884,
0.06744866073131561,
0.02992449700832367,
-0.04893967881798744,
0.010348238982260227,
-0.05320294573903084,
-0.07605081796646118,
-0.06598514318466187,
0.05033653974533081,
0.015591786243021488,
0.07323162257671356,
-0.0361393578350544,
-0.03076586127281189,
-0.03499525412917137,
-0.007379529066383839,
-0.003685629926621914,
0.038469355553388596,
-0.050647322088479996,
0.004724058788269758,
-0.04411463439464569,
-0.04004717618227005,
0.05395963042974472,
-0.0064552356489002705,
0.004339681938290596,
-0.032004907727241516,
0.00027017996762879193,
0.08394449204206467,
0.004071416333317757,
0.06523500382900238,
-0.023818617686629295,
0.04291066527366638,
0.06672996282577515,
-0.06820964813232422,
0.020534152165055275,
-0.026766203343868256,
0.06589462608098984,
-0.062671959400177,
-0.08462675660848618,
-0.02623344026505947,
0.07740890234708786,
-0.10994517058134079,
-0.072372667491436,
0.02333628199994564,
-0.026760639622807503,
0.013966573402285576,
-0.0623926967382431,
0.034648630768060684,
0.0026888621505349874,
0.03713204711675644,
0.037270739674568176,
-0.013372892513871193,
-0.025150151923298836,
0.048390526324510574,
-0.05779869481921196,
0.04069982096552849
] |
12
Chapter 1
1.5 Surds
If n is an integer that is not a square number, then any multiple of ββ―__
n is called a surd.
Examples of surds are ββ―__
2 , ββ―___ 19 and 5 ββ―__
2 .
Surds are examples of irrational numbers.
The decimal expansion of a surd is never-ending and never repeats, for example
ββ―__
2 = 1.414213562...
You can use surds to write exact answers to calculations.
β You can manipulate surds using these rules:
β’ ββ―___ ab = ββ―__
a Γ ββ―__
b
β’ ββ―__
a __ b = ββ―__
a ___
ββ―__
b Irr ational numbers cannot be written
in the form a __ b where a and b are integers.
Surds are examples of irrational numbers .Notation
Simplify:
a ββ―___ 12 b ββ―___ 20 ____ 2 c 5 ββ―__
6 β 2 ββ―___ 24 + ββ―____ 294 Example 12
a ββ―___ 12 = ββ―_______ (4 Γ 3)
= ββ―__
4 Γ ββ―__
3 = 2 ββ―__
3
b ββ―___ 20 ____ 2 = ββ―__
4 Γ ββ―__
5 ________ 2
= 2 Γ ββ―__
5 _______ 2 = ββ―__
5
c 5 ββ―__
6 β 2 ββ―___ 24 + ββ―_____ 294
= 5 ββ―__
6 β 2 ββ―__
6 ββ―__
4 + ββ―__
6 Γ ββ―___ 49
= ββ―__
6 (5 β 2 ββ―__
4 + ββ―___ 49 )
= ββ―__
6 (5 β 2 Γ 2 + 7)
= ββ―__
6 (8)
= 8 ββ―__
6 ββ―__
6 is a common factor. ββ―__
4 = 2Look for a factor of 12 that is a square number.
Use the rule ββ―___ ab = ββ―__
a Γ ββ―__
b . ββ―__
4 = 2
ββ―___ 20 = ββ―__
4 Γ ββ―__
5
Cancel by 2.
Work out the square roots ββ―__
4 and ββ―___ 49 .
5 β 4 + 7 = 8
|
[
-0.01012262050062418,
0.0574290007352829,
0.003593538887798786,
0.008551051840186119,
-0.011814025230705738,
-0.016166184097528458,
0.0020741401240229607,
0.0040824878960847855,
0.0719275176525116,
-0.01108666229993105,
0.014886484481394291,
-0.009672768414020538,
0.09430824220180511,
-0.02795572578907013,
-0.003991324454545975,
-0.02511875331401825,
-0.010585276409983635,
0.1058078408241272,
0.0034842821769416332,
0.02685193158686161,
0.11780902743339539,
0.03525006026029587,
0.05099304020404816,
0.0014041984686627984,
-0.002995789051055908,
0.02848147414624691,
0.03236325457692146,
-0.03939184546470642,
-0.033571742475032806,
-0.058084819465875626,
0.00774878915399313,
0.08618885278701782,
0.0953216403722763,
-0.010456262156367302,
0.018808638677001,
-0.074954554438591,
0.05281819775700569,
-0.002009266521781683,
-0.10857367515563965,
-0.05289478227496147,
0.02357230894267559,
0.008350743912160397,
0.0016889921389520168,
-0.011521751992404461,
0.025115033611655235,
-0.027049679309129715,
-0.09489360451698303,
0.030654331669211388,
0.025273796170949936,
0.007758366409689188,
0.009907600469887257,
0.009230358526110649,
-0.07354176789522171,
-0.0271490179002285,
-0.0013959119096398354,
-0.01810169778764248,
-0.07996727526187897,
-0.03170562908053398,
-0.025869322940707207,
0.009884786792099476,
0.010463859885931015,
-0.009203672409057617,
-0.0008325095404870808,
-0.0058309403248131275,
-0.04449044167995453,
-0.0073823826387524605,
0.001776455552317202,
0.0034385130275040865,
0.03569094091653824,
0.032639000564813614,
0.02090226113796234,
0.014550378546118736,
-0.04726273939013481,
-0.026637405157089233,
0.039910733699798584,
-0.06152768433094025,
-0.08884258568286896,
0.003981919027864933,
0.007242782507091761,
-0.02871062234044075,
-0.07449856400489807,
-0.0024793390184640884,
0.04985494539141655,
-0.039184655994176865,
0.018382418900728226,
-0.06318958103656769,
0.13430353999137878,
0.007860803045332432,
0.02391677349805832,
-0.07127348333597183,
0.06243652105331421,
-0.030383024364709854,
0.023299405351281166,
-0.03855620324611664,
-0.018560189753770828,
0.025762736797332764,
-0.010711350478231907,
-0.06793218851089478,
0.010704767890274525,
-0.011191846802830696,
0.07550104707479477,
-0.07977548241615295,
-0.04506346210837364,
0.012606063857674599,
-0.09091472625732422,
-0.07903695106506348,
-0.0377228744328022,
-0.05027605965733528,
0.08825384080410004,
-0.0998382642865181,
-0.03064517304301262,
0.028085680678486824,
-0.0033078743144869804,
-0.017092853784561157,
-0.008943438529968262,
-0.12940071523189545,
0.05369609594345093,
-0.04078807681798935,
-0.031305331736803055,
-0.04006338492035866,
0.06561360508203506,
0.019195236265659332,
0.05808926746249199,
0.027802040800452232,
-0.028518233448266983,
0.02835209295153618,
0.027302537113428116,
-0.006139271892607212,
-0.07067814469337463,
-0.005421312525868416,
0.005946509074419737,
-0.039787061512470245,
-0.09393522143363953,
0.050750091671943665,
-0.06258851289749146,
-0.07368889451026917,
0.011459100060164928,
0.057039421051740646,
-0.07529715448617935,
0.039233170449733734,
0.018539924174547195,
0.018128057941794395,
0.04744064807891846,
-0.04238572716712952,
0.04996059462428093,
0.04677168279886246,
0.014867119491100311,
0.04473065584897995,
-0.030590418726205826,
0.052547529339790344,
-0.016822846606373787,
0.03395823389291763,
-0.015467185527086258,
0.039589401334524155,
-0.010906976647675037,
0.03726436570286751,
0.06463388353586197,
0.012212986126542091,
-0.011040819808840752,
-0.002010005060583353,
-0.002608079230412841,
0.018368566408753395,
-0.01435556448996067,
0.05117195099592209,
0.065340057015419,
0.05116220936179161,
0.01404450461268425,
-0.0006030331715010107,
0.03795899823307991,
-0.05056114122271538,
0.04958212375640869,
0.038132444024086,
0.12951645255088806,
0.01830855943262577,
-0.03357524797320366,
-0.04266287386417389,
0.047483962029218674,
-0.023438598960638046,
0.013657416217029095,
-0.0668581873178482,
-0.003753358731046319,
-0.010776967741549015,
0.028355523943901062,
-0.036540862172842026,
0.0072320448234677315,
-0.08041837811470032,
-0.12199362367391586,
0.060966573655605316,
0.0284140445291996,
0.0006422103033401072,
-0.07399974763393402,
-0.06835002452135086,
-0.11762066930532455,
-0.07224225997924805,
0.024482902139425278,
-0.0482458621263504,
0.027325088158249855,
0.016476698219776154,
-0.10758204013109207,
0.05843473970890045,
-0.0014086022274568677,
-0.0032209870405495167,
0.023618683218955994,
-0.08272331207990646,
0.06833665817975998,
0.06262286007404327,
0.03663535788655281,
0.04083551466464996,
0.07959359139204025,
0.06469396501779556,
-0.08136119693517685,
-0.053341660648584366,
-0.03990296646952629,
-0.04076598584651947,
-0.056061409413814545,
-0.02424723468720913,
-0.03886226937174797,
-0.06253819167613983,
-0.05051666498184204,
-0.08546329289674759,
-0.00897926464676857,
-0.026965992525219917,
0.0051768459379673,
5.544095311404498e-33,
-0.03695519268512726,
0.04330996051430702,
-0.11433914303779602,
0.032759878784418106,
-0.05108512192964554,
-0.01364745944738388,
0.0033641967456787825,
-0.008814315311610699,
-0.004399167839437723,
-0.04533236101269722,
-0.0884595662355423,
0.039959896355867386,
0.05098296329379082,
0.04161868989467621,
-0.12325140088796616,
-0.020292360335588455,
-0.08475302904844284,
-0.09306374192237854,
0.011505370028316975,
-0.006994657218456268,
-0.03116121143102646,
0.03977713733911514,
0.021246355026960373,
-0.00838736817240715,
0.030136944726109505,
0.09969528019428253,
-0.03775184974074364,
0.00884962547570467,
0.041173581033945084,
0.04134141653776169,
-0.03709098696708679,
0.006718655116856098,
0.06202638894319534,
-0.01718221604824066,
-0.07212968915700912,
-0.040959786623716354,
0.037499360740184784,
-0.030329493805766106,
-0.06142957881093025,
0.012567401863634586,
0.1104980930685997,
0.002687958301976323,
0.07325731962919235,
0.004695117007941008,
-0.023964159190654755,
0.0006913017714396119,
0.027579624205827713,
0.09677460044622421,
-0.03640212491154671,
-0.07592184096574783,
-0.038571033626794815,
-0.025013821199536324,
0.04051967337727547,
0.09260444343090057,
-0.008185319602489471,
0.005073161330074072,
0.03491530194878578,
0.0038100716192275286,
-0.009131224825978279,
0.0026395197492092848,
-0.05882072448730469,
-0.007065603509545326,
0.06473934650421143,
0.08179427683353424,
-0.0099473986774683,
-0.03888629376888275,
-0.005160361994057894,
0.08241866528987885,
-0.004737021867185831,
0.104163259267807,
0.02148360013961792,
0.1300165057182312,
-0.09670685231685638,
-0.08702610433101654,
-0.033926405012607574,
-0.005052290856838226,
-0.054025616496801376,
0.08127003908157349,
0.023867156356573105,
-0.061949457973241806,
-0.06581404060125351,
-0.027217021211981773,
-0.04003831371665001,
-0.016216836869716644,
-0.11503132432699203,
-0.04589541628956795,
0.06956420093774796,
0.035462867468595505,
-0.0287800170481205,
-0.02457522600889206,
0.007589911576360464,
0.041780758649110794,
-0.03545909374952316,
0.02807842381298542,
-0.01669510267674923,
7.402593389698798e-32,
0.016304101794958115,
0.009502258151769638,
0.016064278781414032,
-0.034392572939395905,
0.06310343742370605,
0.05730034038424492,
0.023833725601434708,
0.040731459856033325,
0.012269902974367142,
-0.04474994167685509,
0.05113264545798302,
0.07121609896421432,
-0.020008955150842667,
0.013270560652017593,
-0.01600515842437744,
-0.06648308038711548,
0.08772013336420059,
0.008380988612771034,
-0.011297615244984627,
0.1091303750872612,
0.0975792407989502,
-0.010134144686162472,
-0.08626686036586761,
0.0437481626868248,
-0.01069820486009121,
0.03576980158686638,
-0.051817040890455246,
-0.021362408995628357,
-0.00330104841850698,
-0.04911130294203758,
-0.003673231927677989,
0.007473895791918039,
0.045465581119060516,
-0.01287915837019682,
0.0795760452747345,
0.07293187826871872,
0.07755403965711594,
0.07315489649772644,
-0.06460554897785187,
0.06032610684633255,
-0.03836539760231972,
-0.07946562767028809,
-0.044775474816560745,
0.11827120184898376,
0.05315684899687767,
-0.12234478443861008,
-0.06546833366155624,
-0.03089357726275921,
-0.006811572704464197,
0.012933056801557541,
-0.043725594878196716,
-0.015640640631318092,
-0.05475793406367302,
0.013725705444812775,
-0.008206884376704693,
0.008915058337152004,
0.03657235950231552,
0.020794522017240524,
0.010069028474390507,
-0.019487792626023293,
-0.04506360739469528,
0.10224620997905731,
-0.0069566406309604645,
0.1539362519979477
] |
13Algebraic expressions
Expand and simplify if possible:
a ββ―__
2 (5 β ββ―__
3 ) b (2 β ββ―__
3 )(5 + ββ―__
3 ) Example 13
a ββ―__
2 (5 β ββ―__
3 )
= 5 ββ―__
2 β ββ―__
2 ββ―__
3
= 5 ββ―__
2 β ββ―__
6
b (2 β
ββ―__
3 )(5 + ββ―__
3 )
= 2(5
+ ββ―__
3 ) β ββ―__
3 (5 + ββ―__
3 )
= 10
+ 2 ββ―__
3 β 5 ββ―__
3 β ββ―__
9
= 7 β
3 ββ―__
3 β __
2 Γ 5 β β __
2 Γ β __
3
Using ββ―__
a Γ β __
b = β ____ ab
Collect like terms: 2 β __
3 β 5 β __
3 = β3 β __
3
Simplify any roots if possible: β __
9 = 3
1 Do not use your calcula tor for this exercise. Simplify:
a ββ―___ 28 b ββ―___ 72 c ββ―___ 50
d ββ―__ 32 e ββ―__ 90 f ββ―__ 12 ____ 2
g ββ―___ 27 ____ 3 h ββ―__ 20 + ββ―__ 80 i ββ―___ 200 + ββ―__ 18 β ββ―__ 72
j ββ―___ 175 + ββ―__ 63 + 2 ββ―__ 28 k ββ―__ 28 β 2 ββ―__ 63 + ββ―__
7 l ββ―__ 80 β 2 ββ―__ 20 + 3 ββ―__ 45
m 3 ββ―___ 80 β 2 ββ―___ 20 + 5 ββ―___ 45 n ββ―___ 44 ____ ββ―___ 11 o ββ―___ 12 + 3 ββ―___ 48 + ββ―___ 75
2 Expand and simplify if possible:
a β __
3 (2 + β __
3 ) b β __
5 (3 β β __
3 ) c β __
2 (4 β β __
5 )
d (2 β β __
2 )(3 + β __
5 ) e (2 β β __
3 )(3 β β __
7 ) f (4 + β __
5 )(2 + β __
5 )
g (5 β β __
3 )(1 β β __
3 ) h (4 + β __
3 )(2 β β __
3 ) i (7 β β ___ 11 )(2 + β ___ 11 )
3 Simplify β ___ 75 β β ___ 12 giving your answer in the form a β __
3 , where a is an integer. (2 marks) EExercise 1E
1.6 Rationalising denominators
If a fraction has a surd in the denominator, it is sometimes useful to rearrange it so that the
denominator is a rational number. This is called rationalising the denominator.
β The rules to rationalise denominators are:
β’For fractions in the f
orm 1 ___
ββ―__
a , multiply the numerator and denominat
or by ββ―__
a .
β’For fractions in the f
orm 1 ______
a + ββ―__
b , multiply the numerator and denominat
or by a β ββ―__
b .
β’For fractions in the f
orm 1 ______
a β ββ―__
b , multiply the numerator and denominat
or by a + ββ―__
b .Expand the brackets completely before you simplify.
|
[
-0.04168540984392166,
0.13059596717357635,
0.04445458948612213,
-0.08914688229560852,
-0.027635483071208,
0.0023401714861392975,
0.019677739590406418,
-0.11952056735754013,
-0.013674577698111534,
-0.045642755925655365,
-0.07663243263959885,
-0.014435633085668087,
0.005501685198396444,
-0.006693956442177296,
0.07538321614265442,
0.025151744484901428,
-0.003913607448339462,
0.08426861464977264,
-0.1222277358174324,
-0.05891985446214676,
0.018249636515975,
-0.06277886033058167,
0.06511729210615158,
0.07082895934581757,
0.0004270474018994719,
0.02124612219631672,
-0.037578679621219635,
0.012849707156419754,
0.021405324339866638,
-0.04180707782506943,
0.013346430845558643,
-0.025358298793435097,
0.1096821129322052,
-0.12727026641368866,
0.032965805381536484,
-0.03825753182172775,
0.015418943017721176,
0.04261057823896408,
-0.017110414803028107,
-0.009704836644232273,
0.01600579358637333,
0.014391746371984482,
0.04748217761516571,
0.017073744907975197,
-0.07582147419452667,
-0.024616267532110214,
0.04502519965171814,
0.01931924559175968,
-0.04241291806101799,
-0.014722613617777824,
0.0003243915271013975,
-0.010698795318603516,
-0.09318622201681137,
0.04326246678829193,
0.016291052103042603,
-0.08382253348827362,
-0.01784556731581688,
0.06064469367265701,
-0.01871490851044655,
-0.012838585302233696,
0.0010059475898742676,
0.037433333694934845,
0.008870193734765053,
0.01274879276752472,
-0.05189552158117294,
0.011661406606435776,
-0.016874704509973526,
-0.009036746807396412,
-0.03925356641411781,
0.07847806811332703,
0.036204371601343155,
0.03633498027920723,
-0.05394699424505234,
-0.006039887201040983,
0.05372710898518562,
0.0026548246387392282,
-0.09753010421991348,
-0.08273929357528687,
-0.007000611163675785,
-0.025778956711292267,
-0.06144740805029869,
-0.032331544905900955,
0.04356240853667259,
0.03542261943221092,
0.0284136775881052,
0.005839932709932327,
0.04606116563081741,
0.06247807294130325,
0.04385706037282944,
-0.0366390235722065,
0.027524782344698906,
-0.027993351221084595,
-0.055425599217414856,
-0.09060420095920563,
-0.05222640186548233,
-0.039509713649749756,
-0.02250603958964348,
-0.06405983120203018,
0.007669622544199228,
0.07394593209028244,
0.09009826183319092,
-0.014316115528345108,
-0.009732670150697231,
-0.02311686798930168,
-0.08541939407587051,
-0.056850865483284,
-0.00930753629654646,
-0.014532318338751793,
0.103592649102211,
-0.034281305968761444,
-0.10240677744150162,
0.03412109240889549,
0.038124583661556244,
-0.02533365599811077,
-0.013364775106310844,
0.010967682115733624,
0.042155370116233826,
0.017030948773026466,
0.06529267132282257,
-0.005313871894031763,
0.0037061944603919983,
-0.033865757286548615,
0.051895465701818466,
0.03285475820302963,
0.021626580506563187,
-0.02620462328195572,
-0.05007392168045044,
0.0583798922598362,
-0.014039847999811172,
0.01775074191391468,
-0.017833184450864792,
0.0038540710229426622,
-0.031585995107889175,
0.009697865694761276,
-0.06267081201076508,
-0.13091672956943512,
-0.027798358350992203,
0.07915196567773819,
-0.015323193743824959,
0.03120276890695095,
0.07251043617725372,
0.03394397720694542,
0.05798036605119705,
0.05962201580405235,
-0.018384339287877083,
-0.016657257452607155,
-0.028195954859256744,
0.007704081013798714,
-0.019430741667747498,
0.058045800775289536,
-0.04332411661744118,
0.004074849188327789,
-0.02214827947318554,
0.019684728235006332,
0.04090256243944168,
-0.08590661734342575,
0.0016866441583260894,
-0.027515992522239685,
-0.07433628290891647,
0.07740490138530731,
-0.012326394207775593,
0.026303645223379135,
-0.038351211696863174,
0.07873459905385971,
0.0594724677503109,
0.0629466101527214,
0.04095233604311943,
-0.00612508924677968,
-0.00819474644958973,
0.016532614827156067,
0.012927774339914322,
0.08733048290014267,
0.009690649807453156,
0.00009638669871492311,
-0.02079317905008793,
0.023574860766530037,
0.08706207573413849,
-0.06743917614221573,
-0.04643300175666809,
-0.019813980907201767,
0.07728289067745209,
-0.06113254278898239,
0.0203506201505661,
0.09256813675165176,
-0.07808306068181992,
-0.029548518359661102,
-0.011241401545703411,
0.039808157831430435,
0.0055525717325508595,
-0.07723169028759003,
0.014236819930374622,
-0.11865131556987762,
-0.13125059008598328,
-0.018695443868637085,
-0.009658177383244038,
0.011557025834918022,
-0.05912524461746216,
-0.08306311070919037,
-0.04573296755552292,
0.0766533762216568,
0.005383568815886974,
0.02938651107251644,
0.03775664418935776,
-0.12341482192277908,
0.006269414909183979,
0.03170226141810417,
-0.030451301485300064,
-0.018952084705233574,
0.07240083068609238,
0.0536101795732975,
-0.037379663437604904,
0.014758273959159851,
-0.023075362667441368,
0.0358598530292511,
0.0041407374665141106,
-0.05028914287686348,
0.06542763859033585,
0.006999449338763952,
-0.06451371312141418,
-0.0038497638888657093,
-0.09232185035943985,
-0.020216980949044228,
0.07068636268377304,
-1.9428456754375532e-33,
0.017847763374447823,
0.06421766430139542,
-0.07643791288137436,
-0.021973149850964546,
-0.006487988401204348,
-0.0883718729019165,
0.012623002752661705,
-0.10887201875448227,
-0.013875475153326988,
-0.01999673619866371,
-0.044335246086120605,
0.0017606342444196343,
0.04962063208222389,
-0.036239564418792725,
-0.015010938979685307,
-0.007054303772747517,
-0.007952910847961903,
-0.019633444026112556,
-0.012179802171885967,
-0.03762080892920494,
0.033925607800483704,
0.0209785308688879,
-0.06776051223278046,
-0.006127786822617054,
-0.037955667823553085,
0.10052057355642319,
0.0014189572539180517,
-0.03144204616546631,
0.07184822857379913,
0.06901508569717407,
0.01552114449441433,
-0.04479854181408882,
0.07022999227046967,
0.09359589964151382,
0.013243800029158592,
-0.05971241742372513,
0.05509354546666145,
-0.000464575452497229,
0.06801392883062363,
-0.03906983882188797,
0.04037408158183098,
0.01564757339656353,
0.07044266909360886,
-0.051185477524995804,
0.008087665773928165,
-0.010289791971445084,
0.05558527261018753,
0.11410167813301086,
-0.07169943302869797,
-0.008215761743485928,
-0.06685356050729752,
-0.09204594790935516,
-0.08585833758115768,
0.02029193751513958,
-0.014315714128315449,
0.02570158615708351,
0.09909773617982864,
0.008917993865907192,
0.05126216262578964,
-0.0022561249788850546,
-0.06467190384864807,
-0.05335348844528198,
0.05937807634472847,
-0.024943316355347633,
0.0159470085054636,
0.028377430513501167,
-0.006559303961694241,
0.015409426763653755,
-0.003197422716766596,
0.019910449162125587,
0.031249316409230232,
0.0769612118601799,
-0.14795970916748047,
-0.06986300647258759,
0.011227787472307682,
0.03804076462984085,
0.004679659381508827,
0.07728546112775803,
-0.06463824957609177,
-0.11735385656356812,
-0.03008453920483589,
0.003655735170468688,
-0.0006424466264434159,
-0.056980930268764496,
-0.14248938858509064,
-0.05377107113599777,
0.07746503502130508,
0.022500282153487206,
0.019071387127041817,
-0.005799758248031139,
-0.0007232080097310245,
0.053637079894542694,
0.06292685866355896,
-0.0692695677280426,
0.0169590525329113,
8.113285519262445e-32,
0.03324580937623978,
0.07550198584794998,
-0.04408586770296097,
-0.019670967012643814,
0.06862913817167282,
-0.0176516305655241,
-0.004373163916170597,
-0.014474654570221901,
0.10818309336900711,
-0.09022097289562225,
-0.006445339880883694,
0.03305015340447426,
-0.001164677320048213,
-0.0008136873948387802,
-0.03745144605636597,
0.004357048310339451,
0.01734052039682865,
-0.0030096727423369884,
0.01967591978609562,
0.05620322749018669,
-0.03887731954455376,
0.008841289207339287,
-0.04452257230877876,
0.007603051606565714,
-0.007705165073275566,
0.016968324780464172,
-0.09747055172920227,
0.038462404161691666,
-0.016737408936023712,
-0.0742771327495575,
0.03204852342605591,
0.04149119555950165,
0.018354417756199837,
-0.010313543491065502,
-0.04583403095602989,
0.037482086569070816,
0.06361500173807144,
0.06103121489286423,
-0.013645295053720474,
0.0830138623714447,
0.014632661826908588,
-0.0243076104670763,
-0.0335199199616909,
0.02623569406569004,
0.08552168309688568,
-0.07831187546253204,
-0.08841637521982193,
-0.04122216999530792,
0.04495146498084068,
-0.019265221431851387,
-0.00295329000800848,
0.03431374579668045,
-0.0256604366004467,
0.02539665810763836,
-0.12363195419311523,
-0.006728194188326597,
-0.01762893795967102,
0.05621612071990967,
0.03232491388916969,
-0.010075145401060581,
-0.02286280319094658,
0.059940535575151443,
-0.01399141550064087,
0.08001331984996796
] |
14
Chapter 1
Rationalise the denominator of:
a 1 ___ ββ―__
3 b 1 ______ 3 + ββ―__
2 c ββ―__
5 + ββ―__
2 _______ ββ―__
5 β ββ―__
2 d 1 ________ (1 β ββ―__
3 )2 Example 14
a 1 ___ ββ―__
3 = 1 Γ ββ―__
3 ________ ββ―__
3 Γ ββ―__
3
= ββ―__
3 ___ 3
b 1 _______ 3 + ββ―__
2 = 1 Γ (3
β ββ―__
2 ) ________________ (3 + ββ―__
2 )(3 β ββ―__
2 )
= 3 β ββ―__
2 ___________________ 9 β 3 ββ―__
2 + 3 ββ―__
2 β 2
= 3 β ββ―__
2 _______ 7
c ββ―__
5 + ββ―__
2 ________ ββ―__
5 β ββ―__
2 = ( ββ―__
5 + ββ―__
2 )( ββ―__
5 + ββ―__
2 ) __________________ ( ββ―__
5 β ββ―__
2 )( ββ―__
5 + ββ―__
2 )
= 5 + ββ―__
5 ββ―__
2 + ββ―__
2 ββ―__
5 + 2 _____________________ 5 β 2
= 7 + 2 ββ―___ 10 __________ 3
d 1 _________ (1 β ββ―__
3 ) 2 = 1 ________________ (1 β ββ―__
3 )(1 β ββ―__
3 )
= 1 __________________ 1 β ββ―__
3 β ββ―__
3 + ββ―__
9
= 1 ________ 4 β 2 ββ―__
3
= 1 Γ
(4 + 2 ββ―__
3 ) __________________ (4 β 2 ββ―__
3 )(4 + 2 ββ―__
3 )
= 4 +
2 ββ―__
3 ______________________ 16 + 8 ββ―__
3 β 8 ββ―__
3 β 12
= 4 +
2 ββ―__
3 ________ 4 = 2 +
ββ―__
3 _______ 2 Expand the brackets.
β __
3 Γ β __
3 = 3 ββ―__
3 Γ ββ―__
3 = ( ββ―__
3 )2 = 3Multiply the numerator and denominator by ββ―__
3 .
Multiply numerator and denominator by (3 β ββ―__
2 ) .
ββ―__
2 Γ ββ―__
2 = 2
9 β 2 = 7, β3 ββ―__
2 + 3 ββ―__
2 = 0
ββ―__
5 ββ―__
2 = ββ―___ 10 Multiply numerator and denominator by ββ―__
5 + ββ―__
2 .
β ββ―__
2 ββ―__
5 and ββ―__
5 ββ―__
2 cancel each other out.
Simplify and collect like terms. β __
9 = 3
Multiply the numerator and denominator by
4 + 2 β __
3 .
16 β 12 = 4, 8 β __
3 β 8 β __
3 = 0
|
[
-0.014686287380754948,
0.10219313204288483,
0.02072177082300186,
-0.03921493515372276,
0.04777711257338524,
0.019795779138803482,
0.0226100105792284,
0.019607234746217728,
-0.04033176228404045,
0.0047375792637467384,
-0.02863422967493534,
-0.06672503054141998,
0.012484178878366947,
-0.03344329074025154,
-0.002536995569244027,
0.002237375359982252,
-0.03598521277308464,
0.07347287982702255,
-0.0749993845820427,
0.05480792745947838,
0.04840908572077751,
-0.052118465304374695,
-0.01812463253736496,
0.043677423149347305,
0.03763272613286972,
0.007640998810529709,
0.004387952387332916,
-0.023421911522746086,
-0.02583298832178116,
-0.10489501804113388,
0.04917376488447189,
0.07610511034727097,
0.1242225393652916,
-0.06472369283437729,
0.048453234136104584,
-0.07566385716199875,
0.07434823364019394,
0.06085273250937462,
-0.00429823761805892,
0.023646164685487747,
0.010483630932867527,
0.033178407698869705,
-0.006879973225295544,
0.03975771367549896,
-0.010990064591169357,
-0.00563534302636981,
0.01623491570353508,
0.015489486046135426,
0.019696546718478203,
-0.010262195020914078,
-0.024260733276605606,
0.012877938337624073,
-0.12789687514305115,
-0.04316046088933945,
-0.013173717074096203,
-0.040524374693632126,
-0.0235226359218359,
0.043002352118492126,
-0.05070868879556656,
0.023900369182229042,
-0.003405718132853508,
0.028790321201086044,
0.002200267743319273,
0.033087436109781265,
0.018660597503185272,
0.02817455492913723,
0.008074134588241577,
-0.03292001783847809,
-0.024900754913687706,
0.10584213584661484,
-0.05584465712308884,
0.04327539727091789,
-0.03353423625230789,
-0.01214350014925003,
0.13255932927131653,
-0.04801217466592789,
-0.07307859510183334,
-0.08358682692050934,
0.008894051425158978,
-0.03916110098361969,
-0.06633860617876053,
0.03193516284227371,
0.1041748896241188,
-0.03355387970805168,
-0.007048874162137508,
-0.04697389528155327,
0.11726737022399902,
0.017796073108911514,
0.05358271673321724,
-0.04485158622264862,
0.05603746697306633,
-0.08667810261249542,
-0.01981797255575657,
-0.08835501968860626,
-0.055855363607406616,
0.0005134299281053245,
-0.05023084208369255,
-0.032712262123823166,
0.09191528707742691,
0.04031957685947418,
0.06621939688920975,
-0.02495502308011055,
-0.004801414906978607,
0.00974663719534874,
-0.08267573267221451,
-0.06877751648426056,
0.011721819639205933,
-0.07652706652879715,
0.18138514459133148,
-0.027731366455554962,
-0.036184731870889664,
-0.023561527952551842,
0.01363924890756607,
-0.006091584917157888,
0.011366354301571846,
-0.0709303542971611,
0.07230084389448166,
-0.024258341640233994,
0.03996140509843826,
-0.011775673367083073,
0.06472288817167282,
0.015622314997017384,
0.05905983969569206,
-0.00047814808203838766,
0.01606251671910286,
0.035617291927337646,
-0.019550351426005363,
0.022506659850478172,
-0.05814434215426445,
-0.048306576907634735,
-0.01961250603199005,
0.02891545370221138,
-0.080354705452919,
-0.01456828135997057,
-0.04255502298474312,
-0.10008269548416138,
-0.006874865386635065,
0.08189195394515991,
-0.01593758538365364,
-0.030000055208802223,
0.039657093584537506,
0.055382851511240005,
0.039213698357343674,
-0.0007616596412844956,
-0.05589707940816879,
-0.06736090779304504,
-0.05011419951915741,
0.004653130657970905,
0.0014762867940589786,
0.04253976047039032,
-0.006111635360866785,
0.1019899845123291,
-0.0233793742954731,
0.03990505635738373,
-0.024581536650657654,
0.06525292247533798,
0.0889320969581604,
-0.04441007971763611,
-0.006947228219360113,
0.07472460716962814,
-0.051405396312475204,
0.029574256390333176,
-0.05622795969247818,
0.05367520824074745,
0.04780849441885948,
0.04009389504790306,
0.018490951508283615,
0.0974326804280281,
-0.015485582873225212,
-0.023313645273447037,
0.03252117335796356,
-0.0382254533469677,
0.07709922641515732,
0.009793723933398724,
-0.030680978670716286,
0.0026222090236842632,
0.05549238994717598,
0.04564133659005165,
-0.019720297306776047,
0.017002897337079048,
0.044275540858507156,
0.009073436260223389,
0.02441120333969593,
0.03561680018901825,
-0.014614559710025787,
-0.028700806200504303,
-0.06723971664905548,
0.08448269963264465,
0.007855094969272614,
0.005846438929438591,
0.024873916059732437,
-0.03964567184448242,
-0.05255362018942833,
0.01874624192714691,
-0.08358249813318253,
-0.025405967608094215,
-0.02113858237862587,
-0.03871317207813263,
-0.11032794415950775,
0.0770454853773117,
-0.05857248231768608,
0.06384016573429108,
-0.026593349874019623,
-0.11192279309034348,
-0.006166854873299599,
0.03271036967635155,
-0.006677908357232809,
-0.021056894212961197,
-0.00164507154840976,
0.037360552698373795,
-0.003944821655750275,
-0.010380197316408157,
-0.08620161563158035,
-0.10391461849212646,
-0.0007504242821596563,
-0.0197439044713974,
0.035510942339897156,
0.028763987123966217,
-0.05880921334028244,
0.024249088019132614,
0.009829584509134293,
-0.04227997735142708,
0.06732840090990067,
1.1698219293639976e-32,
-0.02104184962809086,
0.014774131588637829,
-0.12717701494693756,
-0.0542556494474411,
-0.02319074608385563,
-0.017074327915906906,
0.026398373767733574,
-0.0753815695643425,
0.012486131861805916,
-0.04339860752224922,
-0.07121050357818604,
0.05368232727050781,
0.022458191961050034,
-0.011320791207253933,
-0.04200424253940582,
-0.02990727312862873,
-0.041250161826610565,
0.006182999350130558,
-0.026568179950118065,
0.005060507915914059,
-0.011279948987066746,
0.012182792648673058,
0.025298066437244415,
-0.0242669265717268,
-0.0495537705719471,
0.08411374688148499,
0.012842081487178802,
-0.04809651896357536,
0.11604128777980804,
0.14862371981143951,
0.009622485376894474,
-0.020756909623742104,
0.03691013902425766,
0.06304118037223816,
-0.0063239699229598045,
-0.12963585555553436,
0.011471247300505638,
-0.02011406235396862,
-0.012869482859969139,
0.0010215335059911013,
0.1006791964173317,
-0.02988305129110813,
0.08020345121622086,
-0.009056485258042812,
0.06801682710647583,
-0.06538796424865723,
0.04731282964348793,
0.06042148172855377,
0.0056096771731972694,
-0.07682844996452332,
-0.07936152070760727,
-0.03972742334008217,
0.0045767901465296745,
0.03534174710512161,
-0.07454769313335419,
-0.006231605540961027,
0.0341293141245842,
-0.046169206500053406,
0.10648014396429062,
-0.02377178706228733,
-0.013691103085875511,
0.010504278354346752,
0.02554321475327015,
0.08545096218585968,
-0.004470094572752714,
-0.014472924172878265,
-0.017952386289834976,
0.00777823943644762,
-0.000877240439876914,
0.010166549123823643,
0.03217875957489014,
0.10385213047266006,
-0.04316442087292671,
-0.03357850760221481,
-0.013732753694057465,
0.05251976102590561,
-0.021014397963881493,
0.05078348517417908,
-0.043503060936927795,
-0.042189959436655045,
-0.07869654148817062,
-0.014225981198251247,
-0.02639334462583065,
-0.04896341264247894,
-0.06908290833234787,
-0.07862991094589233,
0.09553007036447525,
0.08246729522943497,
-0.01617085002362728,
-0.050127383321523666,
-0.031089985743165016,
0.03001832589507103,
0.0016846229555085301,
0.02406192012131214,
-0.029056508094072342,
5.424936404150803e-32,
0.009198829531669617,
-0.06846233457326889,
-0.04877648502588272,
0.014438841491937637,
0.06319109350442886,
0.0208149254322052,
0.01743680238723755,
-0.021110329777002335,
0.07884153723716736,
-0.022955119609832764,
-0.020029617473483086,
0.05177794769406319,
-0.036833446472883224,
0.07880335301160812,
-0.04701840132474899,
-0.02158551663160324,
0.017252178862690926,
-0.009774819016456604,
-0.03351191431283951,
0.04414526745676994,
0.04981416091322899,
0.018645785748958588,
-0.04263383895158768,
0.005554448813199997,
0.09991755336523056,
0.07856358587741852,
-0.11061453074216843,
0.033071063458919525,
0.035111621022224426,
-0.0657026469707489,
0.0737525224685669,
0.013610092923045158,
0.013545731082558632,
-0.03496305271983147,
0.022321874275803566,
0.010343210771679878,
0.050996776670217514,
0.051180947571992874,
0.010261124931275845,
0.04239816218614578,
0.00030695495661348104,
-0.023647159337997437,
-0.08150579035282135,
0.004867056850343943,
0.035543523728847504,
-0.11234150826931,
-0.06192798912525177,
-0.03875177353620529,
0.002015798119828105,
0.0022989283315837383,
-0.04954617843031883,
0.046135563403367996,
-0.04230223223567009,
-0.048444103449583054,
-0.0263956431299448,
-0.08822241425514221,
0.006030152086168528,
0.011361186392605305,
0.05104469880461693,
0.025282924994826317,
-0.017635272815823555,
0.13046647608280182,
0.007386226672679186,
0.09198972582817078
] |
15Algebraic expressions
1 Simplify:
a 1 ___ ββ―__
5 b 1 ____ ββ―___ 11 c 1 ___ ββ―__
2 d ββ―__
3 ____ ββ―___ 15
e ββ―__ 12 ____ ββ―__ 48 f ββ―__
5 ____ ββ―___ 80 g ββ―___ 12 _____ ββ―____ 156 h ββ―__
7 ____ ββ―___ 63
2 Rationa
lise the denominators and simplify:
a 1 ______ 1 + ββ―__
3 b 1 ______ 2 + ββ―__
5 c 1 ______ 3 β ββ―__
7 d 4 ______ 3 β ββ―__
5 e 1 _______ ββ―__
5 β ββ―__
3
f 3 β ββ―__
2 ______ 4 β ββ―__
5 g 5 ______ 2 + ββ―__
5 h 5 ββ―__
2 _______ ββ―__
8 β ββ―__
7 i 11 _______ 3 + ββ―___ 11 j ββ―__
3 β ββ―__
7 _______ ββ―__
3 + ββ―__
7
k ββ―___ 17 β ββ―___ 11 _________ ββ―___ 17 + ββ―___ 11 l ββ―___ 41 + ββ―___ 29 _________ ββ―___ 41 β ββ―___ 29 m ββ―__
2 β ββ―__
3 _______ ββ―__
3 β ββ―__
2
3 Rationa
lise the denominators and simplify:
a 1 ________ (3 β ββ―__
2 ) 2 b 1 ________ (2 + ββ―__
5 ) 2 c 4 ________ (3 β ββ―__
2 ) 2
d 3 ________ (5 + ββ―__
2 ) 2 e 1 ______________ (5 + β __
2 )(3 β β __
2 ) f 2 ______________ (5 β β __
3 )(2 + β __
3 )
4 Simplify 3 β 2 ββ―__
5 _______ ββ―__
5 β 1 giving your answer in the
form p + q ββ―__
5 , where p and q are rational
numbers. (4 marks)E/P
You can check that your answer is in the correct
form by writing down the values of p and q and checking that they are rational numbers.Problem-solving
1 Simplify:
a y3 Γ y5 b 3x2 Γ 2x5 c (4x2)3 Γ· 2x5 d 4b2 Γ 3b3 Γ b4
2 Expand and simplify if possible:a
(x
+ 3)(x β 5) b (2x
β 7)(3x + 1) c (2x
+ 5)(3x β y + 2)
3 Expand and simplify if possible:a
x(x
+ 4)(x β 1) b (x
+ 2)(x β 3)(x + 7) c (2x
+ 3)(x β 2)(3x β 1)
4 Expand the brackets:a
3(5y
+ 4) b 5x2(3 β 5x + 2x2) c 5x(2 x + 3) β 2x(1 β 3x) d 3x2(1 + 3x) β 2x(3x β2)Exercise 1F
Mixed exercise 1
|
[
-0.025421569123864174,
0.11251389235258102,
0.026645056903362274,
-0.07874953001737595,
-0.002589508192613721,
0.006759926211088896,
0.04971057549118996,
-0.01578047312796116,
-0.04731705039739609,
-0.03618544712662697,
-0.06822872906923294,
-0.07045580446720123,
0.01935826614499092,
-0.0059540411457419395,
0.012330446392297745,
0.04577261582016945,
-0.03355155140161514,
0.03778901323676109,
-0.09715127944946289,
0.019087763503193855,
0.11900590360164642,
-0.044978924095630646,
0.04477053880691528,
0.019634557887911797,
-0.06642407178878784,
-0.006035072263330221,
-0.05777251720428467,
-0.0584847517311573,
-0.07346741110086441,
-0.09288189560174942,
-0.0010865735821425915,
0.0069969124160707,
0.1358916163444519,
-0.09002793580293655,
0.009724576026201248,
-0.0217976663261652,
-0.0023166921455413103,
0.03528445214033127,
0.02549973875284195,
-0.026981130242347717,
-0.013872973620891571,
0.042530931532382965,
0.014388063922524452,
0.06319105625152588,
-0.04203234240412712,
-0.029761873185634613,
0.06714345514774323,
0.01981290988624096,
-0.010396669618785381,
0.004347401671111584,
0.039334315806627274,
-0.021012647077441216,
-0.09710357338190079,
0.017400652170181274,
0.044449977576732635,
-0.07547822594642639,
-0.014397677034139633,
0.01655760407447815,
0.0071222810074687,
0.02016446925699711,
-0.015022077597677708,
0.01805400289595127,
0.04486391693353653,
0.02259884960949421,
-0.059637926518917084,
0.03307119384407997,
0.023876966908574104,
0.02366209588944912,
0.03593022748827934,
0.05363839492201805,
-0.015381928533315659,
0.028567131608724594,
-0.03802823647856712,
-0.03228418156504631,
0.08237183839082718,
-0.030465885996818542,
-0.011583494953811169,
-0.09872829169034958,
-0.01179778017103672,
-0.06861934065818787,
-0.06327468901872635,
-0.06400841474533081,
0.05740133300423622,
-0.05209602415561676,
0.01194223016500473,
-0.005080592818558216,
0.05870415270328522,
0.14466629922389984,
0.014919220469892025,
-0.07147539407014847,
0.03839836269617081,
-0.06281955540180206,
-0.002020477084442973,
-0.04708876833319664,
-0.07031848281621933,
-0.025650927796959877,
-0.0056953104212880135,
-0.06396573036909103,
-0.008002620190382004,
0.022210203111171722,
0.06496912986040115,
-0.04453209415078163,
0.04071643203496933,
-0.0002589086361695081,
-0.09412267804145813,
-0.039453160017728806,
-0.040337368845939636,
0.005540810991078615,
0.11920811235904694,
-0.03918420523405075,
-0.07644660770893097,
0.016974223777651787,
0.00411123177036643,
-0.0768696591258049,
0.036257971078157425,
-0.0007531343144364655,
0.028149602934718132,
-0.06085562705993652,
0.057318296283483505,
0.05902169272303581,
-0.03839828073978424,
0.0278497114777565,
0.044078197330236435,
0.020814383402466774,
0.08702471852302551,
0.010433483868837357,
-0.042977746576070786,
0.08147036284208298,
-0.023036547005176544,
0.00737480353564024,
-0.010162955150008202,
0.025945382192730904,
0.0036404700949788094,
0.021377859637141228,
-0.1038237139582634,
-0.14789290726184845,
0.032662879675626755,
0.07929648458957672,
-0.0014356760075315833,
-0.007712685503065586,
0.029640307649970055,
0.04991726204752922,
0.029464449733495712,
0.036440834403038025,
-0.029241707175970078,
-0.037467729300260544,
-0.06426788121461868,
0.05439293757081032,
0.01052619144320488,
0.012625223957002163,
0.03869776427745819,
0.033422306180000305,
-0.05490675941109657,
0.003313754452392459,
0.0017293869750574231,
0.03895885869860649,
0.06818517297506332,
-0.027602020651102066,
-0.06921329349279404,
0.05199617147445679,
-0.003166703972965479,
-0.030241629108786583,
-0.01191472914069891,
0.042073532938957214,
0.08833902329206467,
0.049472857266664505,
0.060055240988731384,
-0.003057564375922084,
-0.01583966054022312,
0.04310337454080582,
0.0024661365896463394,
0.006031549535691738,
0.07138463109731674,
-0.010921661742031574,
-0.022467654198408127,
-0.008611605502665043,
0.11014222353696823,
0.025981055572628975,
-0.04231581091880798,
-0.00665709562599659,
0.04901716858148575,
-0.08187487721443176,
0.012384290806949139,
0.06502469629049301,
-0.022532379254698753,
0.01996137760579586,
-0.05507775768637657,
0.043864745646715164,
-0.02878592722117901,
-0.07549633830785751,
0.03007175587117672,
-0.04569606855511665,
-0.10998430848121643,
-0.02267056331038475,
-0.006139417644590139,
-0.03342556208372116,
-0.09778892993927002,
-0.03559710085391998,
-0.07404609024524689,
0.053934551775455475,
0.02263391949236393,
-0.0018458312842994928,
-0.017117898911237717,
-0.04780762642621994,
0.09711965918540955,
0.007989607751369476,
-0.059175387024879456,
0.023454705253243446,
-0.019281338900327682,
0.04420679807662964,
0.007752811070531607,
-0.017553720623254776,
-0.012735511176288128,
-0.04242364317178726,
0.0463525615632534,
-0.009454343467950821,
0.05394243448972702,
-0.03885049372911453,
-0.1000935435295105,
-0.02487924136221409,
0.01428897026926279,
-0.03780725598335266,
0.0865057185292244,
5.767592419668324e-33,
0.02730369381606579,
0.07233095169067383,
-0.0980117917060852,
-0.06087452545762062,
-0.006458894349634647,
-0.030670536682009697,
0.025697888806462288,
-0.05228821933269501,
-0.0008896942599676549,
0.006181353237479925,
-0.010493874549865723,
-0.009732210077345371,
0.03680998831987381,
-0.03154975175857544,
-0.07475246489048004,
0.008374580182135105,
-0.047017697244882584,
0.04605115205049515,
-0.05214757099747658,
-0.04907721281051636,
-0.04666416347026825,
0.01891423761844635,
0.05342989042401314,
0.03076469711959362,
-0.0868239775300026,
0.09424585849046707,
0.02183074690401554,
-0.0260887760668993,
0.023136522620916367,
0.08295940607786179,
0.09389867633581161,
-0.07956796139478683,
0.0764392614364624,
0.07297704368829727,
-0.04787871241569519,
-0.09454192221164703,
0.04175553098320961,
-0.0247028861194849,
0.04238615557551384,
-0.014701702632009983,
0.07532437145709991,
-0.01205939520150423,
0.12015732377767563,
-0.026953203603625298,
0.03949056565761566,
-0.0016646800795570016,
0.010534358210861683,
0.05770596116781235,
-0.042916905134916306,
-0.035880014300346375,
-0.08223341405391693,
-0.09445217251777649,
-0.07773617655038834,
-0.062123145908117294,
-0.057391416281461716,
-0.008811489678919315,
0.01883857697248459,
-0.028609463945031166,
0.11464673280715942,
0.02857181616127491,
-0.004063002299517393,
-0.002961210673674941,
0.02373683825135231,
0.07205276191234589,
-0.07460787892341614,
-0.007680769544094801,
0.010635508224368095,
0.052972160279750824,
-0.023066988214850426,
0.028229162096977234,
0.041083164513111115,
0.06997384130954742,
-0.0981665775179863,
-0.009380602277815342,
-0.05107356607913971,
0.02990683540701866,
0.033699776977300644,
0.010638592764735222,
0.03290759027004242,
-0.07052543014287949,
-0.027234015986323357,
0.03675568103790283,
0.021877538412809372,
-0.058618128299713135,
-0.11526127904653549,
-0.08617392927408218,
0.06992420554161072,
-0.0015367764281108975,
0.04786660894751549,
-0.024214133620262146,
0.019987402483820915,
0.06636544317007065,
0.061254531145095825,
-0.008830905891954899,
-0.04471088945865631,
6.175480143196027e-32,
-0.0006476342678070068,
0.013155574910342693,
-0.02116147056221962,
0.014075545594096184,
0.037116337567567825,
-0.005045636091381311,
-0.03867574781179428,
-0.033208634704351425,
0.07691926509141922,
-0.06937842071056366,
0.04803846776485443,
0.05692102387547493,
-0.059996914118528366,
0.055680301040410995,
-0.018547166138887405,
-0.03716300427913666,
-0.0034954180009663105,
0.014454834163188934,
-0.007836811244487762,
0.055268771946430206,
-0.015972331166267395,
0.038782618939876556,
-0.024883126839995384,
0.02166605554521084,
0.032457660883665085,
0.07257044315338135,
-0.11014735698699951,
-0.011506428010761738,
0.009176395833492279,
-0.06411605328321457,
0.05925430729985237,
0.0016310391947627068,
-0.0036741753574460745,
-0.00012153127318015322,
-0.030488066375255585,
0.04416615888476372,
0.029511000961065292,
0.0797293484210968,
-0.032704584300518036,
0.14756083488464355,
-0.01422412134706974,
-0.014420104213058949,
-0.002898449543863535,
-0.01282726600766182,
0.014769800938665867,
-0.10268372297286987,
-0.07109994441270828,
-0.019611962139606476,
0.03459806740283966,
-0.012723162770271301,
-0.06428107619285583,
0.0565866082906723,
-0.033015526831150055,
-0.003043131669983268,
-0.10093723237514496,
0.001064957003109157,
0.03741573914885521,
0.07994980365037918,
0.017760692164301872,
0.03407808765769005,
0.031409140676259995,
0.11524799466133118,
-0.025455452501773834,
0.03615158796310425
] |
16
Chapter 1
5 Factorise these expr
essions completely:
a 3x2 + 4x b 4y2 + 10y c x2 + xy + xy2 d 8xy2 + 10x2y
6 Factorise:
a x2 + 3x + 2 b 3x2 + 6x c x2 β 2x β 35 d 2x2 β x β 3
e 5x2 β 13x β 6 f 6 β 5 x β x2
7 Factorise:
a 2x3 + 6x b x3 β 36x c 2x3 + 7x2 β 15x
8 Simplify:
a 9x3 Γ· 3xβ3 b ( 4 3 _ 2 ) 1 _ 3 c 3xβ2 Γ 2x4 d 3 x 1 _ 3 Γ· 6 x 2 _ 3
9 Eva
luate:
a ( 8 ___ 27 ) 2 _ 3
b ( 225 ____ 289 ) 3 _ 2
10 Simplify:
a 3 ____ ββ―___ 63 b ββ―__ 20 + 2 ββ―__ 45 β ββ―__ 80
11 a Find the value of
35x2 + 2x β 48 when x = 25.
b By factorising the expression, sho
w that your answer to part a can be written as the product
of two prime factors.
12 Expand and simplify if possible:
a ββ―__
2 (3 + ββ―__
5 ) b (2 β ββ―__
5 )(5 + ββ―__
3 ) c (6 β ββ―__
2 )(4 β ββ―__
7 )
13 Rationa
lise the denominator and simplify:
a 1 ____ ββ―___
3 b 1 ______ ββ―__
2 β 1 c 3 ______ ββ―__
3 β 2 d ββ―___ 23 β ββ―___ 37 _________ ββ―___ 23 + ββ―___ 37 e 1 ________ (2 + ββ―__
3 )2 f 1 ________ (4 β ββ―__
7 )2
14 a Given tha
t x3 β x2 β 17x β 15 = (x + 3)(x2 + bx + c), where b and c are constants, work out
the values of b and c.
b Hence, fully factorise
x3 β x2 β 17x β 15.
15 Given tha
t y = 1 __ 64 x 3 express each of the following in the form k x n , where k and n are constants.
a y 1 _ 3 (1 mark)
b 4 y β1 (1 mark)
16 Show that 5 _________ ββ―___ 75 β ββ―___ 50 can be written in the form ββ―__
a + ββ―__
b , where a and b are integers. (5 marks)
17 Expand and simplify ( ββ―___ 11 β 5)(5 β ββ―___ 11 ) . (2 marks)
18 Factorise completel
y x β 64 x 3 . (3 marks)
19 Express 27 2x + 1 in the form 3 y , stating y in terms of x. (2 marks)E
E/P
E
E
E/P
|
[
-0.01494672428816557,
0.0855688825249672,
0.011874442920088768,
-0.06033628061413765,
0.04024406522512436,
0.08138574659824371,
-0.03547201678156853,
-0.01816456764936447,
-0.07763617485761642,
0.028328219428658485,
-0.000017930229660123587,
-0.06935379654169083,
0.05119537562131882,
-0.04857812449336052,
-0.017426861450076103,
-0.04517945274710655,
0.005384984891861677,
0.05242493748664856,
-0.08275346457958221,
-0.005995896179229021,
0.051813121885061264,
-0.08918413519859314,
-0.003932260442525148,
-0.07323569059371948,
0.0695558413863182,
-0.04406611993908882,
-0.011512761935591698,
-0.11232160776853561,
-0.026089385151863098,
-0.08059857785701752,
0.05324268713593483,
0.07352038472890854,
0.1040053740143776,
-0.03150314465165138,
0.04474649950861931,
0.018591290339827538,
0.0590326264500618,
-0.015124971978366375,
-0.0005820530350320041,
-0.02269195206463337,
-0.09520217031240463,
0.03163778409361839,
-0.033917635679244995,
0.013815930113196373,
0.018544238060712814,
-0.06107819080352783,
-0.01811785064637661,
0.011955155059695244,
0.04517216235399246,
-0.0869387835264206,
0.04186905920505524,
-0.05420756712555885,
-0.023531269282102585,
0.0714993104338646,
-0.01971179060637951,
-0.030998840928077698,
-0.04821979999542236,
0.04670115187764168,
-0.022346820682287216,
-0.023212719708681107,
-0.0799284279346466,
-0.061781879514455795,
0.010402726009488106,
0.021359557285904884,
-0.004804582800716162,
0.031803615391254425,
-0.005994285922497511,
-0.0930907279253006,
0.04774020239710808,
0.017846470698714256,
-0.08988290280103683,
-0.014913620427250862,
-0.028880232945084572,
-0.042615752667188644,
0.12323713302612305,
0.06356895714998245,
-0.037082232534885406,
-0.031063461676239967,
0.016848232597112656,
-0.0743337869644165,
-0.01959032379090786,
-0.04098577797412872,
0.1170208603143692,
0.0009969357633963227,
0.056623127311468124,
-0.04335125535726547,
0.09934655576944351,
0.011971483007073402,
0.04061726853251457,
-0.066889189183712,
0.02832578495144844,
-0.0984208807349205,
0.020533619448542595,
-0.05211552977561951,
-0.08398827165365219,
-0.06998500227928162,
0.010986790992319584,
-0.09709800779819489,
0.04934539273381233,
0.059045225381851196,
0.027238326147198677,
-0.005090374033898115,
-0.011728083714842796,
-0.04922271519899368,
-0.06755881756544113,
-0.033071376383304596,
0.02125711180269718,
-0.023755641654133797,
0.06649063527584076,
-0.13185396790504456,
-0.10108337551355362,
-0.007154025137424469,
0.00592109514400363,
-0.024580007418990135,
-0.02388552948832512,
-0.06006612256169319,
0.11823724955320358,
-0.05950991064310074,
-0.020600978285074234,
-0.02553486078977585,
0.07010933011770248,
-0.022983305156230927,
0.023057466372847557,
-0.006580126006156206,
0.004712819587439299,
-0.01766338385641575,
0.013011252507567406,
-0.0005716268206015229,
-0.023045387119054794,
-0.04174163192510605,
0.02183305285871029,
0.04686560854315758,
-0.022682474926114082,
0.027033278718590736,
-0.0167431328445673,
-0.09573144465684891,
-0.062346503138542175,
0.08394681662321091,
-0.05346957594156265,
-0.018422655761241913,
-0.005312599241733551,
0.08588407933712006,
0.021086854860186577,
-0.011663512326776981,
-0.0008622509776614606,
-0.0710708424448967,
-0.028336727991700172,
-0.02503061667084694,
0.04828853905200958,
0.08936794102191925,
0.003400583751499653,
-0.030297089368104935,
-0.029410341754555702,
0.029118675738573074,
-0.019236918538808823,
-0.03542928770184517,
-0.00888226367533207,
-0.06411764770746231,
-0.0017138406401500106,
-0.082717664539814,
-0.10866779834032059,
0.012284263968467712,
-0.052778229117393494,
0.07979045808315277,
-0.008449946530163288,
0.022148646414279938,
0.05886426568031311,
0.04993055760860443,
-0.08890928328037262,
0.011048616841435432,
0.016026919707655907,
-0.02131924219429493,
-0.016181515529751778,
0.09964003413915634,
0.016965465620160103,
-0.003674280596897006,
0.061270568519830704,
-0.01755020022392273,
0.0029660742729902267,
0.11433151364326477,
0.04695481061935425,
-0.03081865981221199,
-0.05540953949093819,
0.0531889833509922,
-0.027653655037283897,
0.002702338621020317,
-0.05660523474216461,
0.026997048407793045,
-0.021563611924648285,
-0.041902538388967514,
0.03630455583333969,
-0.06243589147925377,
-0.054468508809804916,
0.051058828830718994,
-0.06530928611755371,
0.0035940995439887047,
0.025174591690301895,
-0.05226787179708481,
-0.12817244231700897,
0.02348150499165058,
0.03884973004460335,
0.000060934089560760185,
0.02020924724638462,
-0.06337396055459976,
0.0013898300239816308,
-0.032154377549886703,
-0.06237783655524254,
-0.033776864409446716,
-0.0029429993592202663,
0.0828598141670227,
0.016806919127702713,
0.059110477566719055,
0.011770525015890598,
-0.030785460025072098,
0.02586827427148819,
0.012020385824143887,
0.10786355286836624,
-0.06253211200237274,
-0.002078306395560503,
-0.09771464765071869,
0.013748090714216232,
-0.11296659708023071,
0.007464576046913862,
-2.3342170421206303e-32,
-0.046714648604393005,
0.012035032734274864,
-0.10369694232940674,
-0.06394470483064651,
0.06068853288888931,
-0.050337765365839005,
0.01881324127316475,
-0.06848954409360886,
0.1391458809375763,
0.0338141992688179,
0.02760881930589676,
-0.014764675870537758,
-0.008905836381018162,
0.01967856101691723,
-0.0796906128525734,
-0.056749552488327026,
-0.04535675421357155,
0.09662356972694397,
0.009996851906180382,
-0.034270599484443665,
-0.02765706181526184,
0.004914224147796631,
-0.007872811518609524,
-0.011461923830211163,
-0.03382093831896782,
-0.023159531876444817,
0.0806742012500763,
-0.08105473965406418,
0.012050710618495941,
0.0418693944811821,
0.019291304051876068,
-0.06442886590957642,
-0.00335947354324162,
0.03477639704942703,
-0.027089213952422142,
-0.10514886677265167,
-0.014416765421628952,
0.022926799952983856,
-0.04604417458176613,
0.022242572158575058,
0.0832420289516449,
-0.024884089827537537,
-0.005962797440588474,
-0.007614393252879381,
0.024786274880170822,
0.050693538039922714,
0.04503354802727699,
0.017311880365014076,
0.03787316009402275,
-0.044047750532627106,
-0.021196747198700905,
-0.008138538338243961,
0.005341831594705582,
0.022566039115190506,
0.02346815913915634,
0.02090374380350113,
0.09842181205749512,
-0.030181793496012688,
0.09066285192966461,
0.021296504884958267,
-0.06897617876529694,
0.02119147777557373,
-0.006489242427051067,
0.026081781834363937,
-0.04704011231660843,
0.01257358305156231,
0.008743691258132458,
0.045943211764097214,
-0.0522259958088398,
0.04565338045358658,
-0.03448688983917236,
0.07637710869312286,
-0.07620376348495483,
0.03596547245979309,
-0.008591965772211552,
0.030980579555034637,
-0.022818874567747116,
0.05756804347038269,
-0.03767832741141319,
-0.009367991238832474,
-0.04218784347176552,
-0.03625807911157608,
0.0004412100533954799,
-0.10248877853155136,
-0.058313433080911636,
-0.023596959188580513,
0.13380080461502075,
0.05943151190876961,
0.05923651158809662,
-0.04402555152773857,
-0.02925930730998516,
-0.0071878209710121155,
0.1399645060300827,
-0.016845740377902985,
0.056901123374700546,
1.4039234660401794e-31,
0.017285563051700592,
-0.008541337214410305,
-0.07981029152870178,
0.006706387270241976,
0.030200757086277008,
0.019822007045149803,
0.016573965549468994,
-0.021094271913170815,
0.0036182289477437735,
-0.057572685182094574,
0.07776185125112534,
0.012235203757882118,
-0.014250651001930237,
0.027887586504220963,
-0.0631854236125946,
-0.018936172127723694,
-0.08907561749219894,
-0.003340210299938917,
-0.027135662734508514,
0.04886828735470772,
-0.019988015294075012,
0.008225684985518456,
0.007088445592671633,
-0.03260600566864014,
0.01870596967637539,
0.07948894798755646,
-0.011004015803337097,
-0.00468668807297945,
0.030346650630235672,
0.020508402958512306,
0.09270656108856201,
-0.09442310780286789,
-0.04202356934547424,
-0.05778668820858002,
-0.034841399639844894,
0.01515924371778965,
0.09998008608818054,
0.015338084660470486,
-0.006899401545524597,
0.03961046412587166,
-0.0011812072480097413,
-0.08855774998664856,
-0.01267204713076353,
-0.018364192917943,
-0.03345491364598274,
-0.08615868538618088,
-0.03451073169708252,
-0.005418781656771898,
0.016235006973147392,
-0.006224939599633217,
-0.06494923681020737,
0.0017967328894883394,
-0.017695631831884384,
-0.010423580184578896,
0.044226761907339096,
-0.06296563148498535,
-0.0367245078086853,
0.03913399204611778,
-0.0016397885046899319,
-0.05129208788275719,
-0.016555149108171463,
0.08437388390302658,
-0.04877398908138275,
0.0283260066062212
] |
17Algebraic expressions
20 Solve the equation 8 + x ββ―___ 12 = 8x ___ ββ―__
3
Give y
our answer in the form a ββ―__
b where a and b are integers. (4 marks)
21 A rectangle has a length of (1 + ββ―__
3 ) cm and area of ββ―___ 12 cm2.
Calculate the width of the rectangle in cm.
Express your answer in the form a + b ββ―__
3 , where a and b are integers to be found.
22 Show that (2 β ββ―__
x ) 2 ________ ββ―__
x can be written as 4 x β 1 _ 2 β 4 + x 1 _ 2 . (2 marks)
23 Given tha
t 243 ββ―__
3 = 3 a , find the value of a. (3 marks)
24 Given tha
t 4 x 3 + x 5 _ 2 ________ ββ―__
x can be written in the form 4 x a + x b , write down the value of a
and the value of b. (2 marks)E/P
P
E
E/P
E/P
1 You can use the laws of indices to simplify powers of the same base.
β am Γ an = am + n β am Γ· an = am β n
β (am)n = amn β (ab )n = anbn
2 Factorising is the opposite of expanding brackets.
3 A quadratic expr
ession has the form ax2 + bx + c where a, b and c are real numbers and a β 0.
4 x2 β y2 = (x + y)(x β y)
5 You can use the law
s of indices with any rational power.
β a 1 __ m = m ββ―__
a β a n __ m = m ββ―__
a n
β a βm = 1 ___ a m β a 0 = 1
6 You can manipulate sur
ds using these rules:
β ββ―____ ab = ββ―__
a Γ ββ―__
b β ββ―__
a __ b = ββ―__
a ___
ββ―__
b
7 The rules to r
ationalise denominators are:
β Fractions in the f
orm 1 ___
ββ―__
a , multiply the numerator and denominat
or by ββ―__
a .
β Fractions in the f
orm 1 ______
a + ββ―__
b , multiply the numerator and denominat
or by a β ββ―__
b .
β Fractions in the f
orm 1 ______
a β ββ―__
b , multiply the numerator and denominat
or by a + ββ―__
b .Summary of key pointsa Simplify ( ββ―__
a + ββ―__
b ) ( ββ―__
a β ββ―__
b ) .
b Hence show that 1 _______
ββ―__
1 + ββ―__
2 + 1 _______
ββ―__
2 + ββ―__
3 + 1 _______
ββ―__
3 + ββ―__
4 + ... +
1 _________
ββ―___ 24 + ββ―___ 25 = 4Challenge
|
[
0.0794546827673912,
0.1015685424208641,
-0.026054993271827698,
-0.03144179284572601,
-0.0501992367208004,
0.028811698779463768,
0.02456670068204403,
0.0014317742316052318,
-0.08592641353607178,
-0.038421571254730225,
-0.026034310460090637,
-0.07003024220466614,
0.0016906982054933906,
-0.03628546744585037,
0.028632545843720436,
-0.009973912499845028,
-0.049276601523160934,
0.03988689184188843,
-0.0874297022819519,
0.03455411270260811,
0.09614512324333191,
-0.06418702751398087,
-0.009941089898347855,
0.04497629776597023,
0.029392071068286896,
-0.00826894212514162,
-0.09375516325235367,
0.019597768783569336,
0.004989894572645426,
-0.019453203305602074,
-0.0019989211577922106,
0.032102007418870926,
0.1746433824300766,
-0.08258923888206482,
0.04748069867491722,
-0.009468890726566315,
0.06370139122009277,
0.046447038650512695,
-0.020059799775481224,
-0.04851274937391281,
-0.06211413815617561,
0.05319362133741379,
0.08008408546447754,
0.03103066235780716,
0.026823444291949272,
-0.1022924929857254,
0.0714820995926857,
0.02256849594414234,
0.022774891927838326,
-0.036679498851299286,
0.06634176522493362,
0.033001858741045,
-0.08854527026414871,
0.04294167086482048,
0.008650161325931549,
-0.051620740443468094,
0.0008335934835486114,
0.056376438587903976,
0.08284572511911392,
0.000453609274700284,
0.00357151054777205,
0.00438486784696579,
0.02220390923321247,
0.0508514866232872,
-0.021972352638840675,
0.044033199548721313,
-0.08234420418739319,
-0.027373678982257843,
0.03660496696829796,
0.03440137580037117,
-0.00826748926192522,
0.041182175278663635,
-0.0666261836886406,
-0.027900196611881256,
0.03526251018047333,
-0.08807436376810074,
-0.09837368130683899,
-0.06689680367708206,
0.04060552269220352,
-0.042698025703430176,
-0.04666877165436745,
0.016829509288072586,
0.09048683941364288,
-0.04474795609712601,
0.025632718577980995,
-0.014841403812170029,
0.037617459893226624,
0.07762773334980011,
-0.015190773643553257,
-0.07197244465351105,
0.04547915607690811,
-0.04227203130722046,
-0.04371783882379532,
-0.01586993783712387,
0.028244102373719215,
-0.07600496709346771,
-0.01737970858812332,
-0.08837734162807465,
-0.008055297657847404,
0.08204839378595352,
0.010962320491671562,
-0.0365837924182415,
0.033402834087610245,
0.0002748290717136115,
-0.019529234617948532,
-0.004829907324165106,
-0.039771873503923416,
-0.04389140009880066,
0.1182483434677124,
-0.04080181196331978,
-0.09941114485263824,
0.02780001610517502,
0.0328252911567688,
0.04502134397625923,
0.03960370272397995,
0.016323598101735115,
0.06717050820589066,
-0.02927410416305065,
0.005019041243940592,
0.019636880606412888,
0.010363508015871048,
-0.0028704481665045023,
0.1232907772064209,
-0.07066072523593903,
0.007141043897718191,
-0.011039728298783302,
-0.01114907767623663,
0.05788355693221092,
-0.004326025024056435,
-0.0704294741153717,
0.0067545161582529545,
-0.02088390477001667,
-0.028484847396612167,
-0.01790832355618477,
-0.011835435405373573,
-0.07800435274839401,
-0.036076389253139496,
0.0840403139591217,
-0.012948313727974892,
-0.041884031146764755,
0.11130031198263168,
0.09308275580406189,
0.11335115879774094,
0.028685977682471275,
-0.011948096565902233,
0.026175493374466896,
0.006265405099838972,
0.006524316500872374,
-0.03279772773385048,
0.05191710218787193,
0.0021581228356808424,
0.013851291500031948,
-0.05454948544502258,
0.013259327970445156,
-0.013278667815029621,
-0.04721932113170624,
-0.0431058406829834,
-0.014102684333920479,
-0.0801975280046463,
0.025253556668758392,
-0.053074490278959274,
0.07014886289834976,
0.02152363583445549,
0.02543352171778679,
0.08458089828491211,
0.027454650029540062,
0.008427736349403858,
-0.01065277773886919,
0.004757861606776714,
0.03099936619400978,
-0.0074784099124372005,
0.06628984957933426,
0.05446223169565201,
0.05958108231425285,
-0.0026213997043669224,
-0.0776466503739357,
0.09456190466880798,
-0.06447666883468628,
0.012138276360929012,
0.04194820672273636,
0.10250332951545715,
-0.07829754054546356,
-0.040542036294937134,
0.015134471468627453,
-0.010112832300364971,
-0.019033070653676987,
-0.01992330327630043,
0.07890736311674118,
-0.08627793937921524,
-0.019869301468133926,
0.026181744411587715,
-0.08663368970155716,
-0.08547224849462509,
0.030887387692928314,
-0.00950618740171194,
-0.02825331874191761,
0.01884051226079464,
-0.07497862726449966,
-0.11840266734361649,
0.04104985296726227,
-0.006140063516795635,
0.006897883955389261,
-0.016947917640209198,
-0.1296284794807434,
0.07054198533296585,
0.07487908005714417,
-0.016238760203123093,
0.01023958157747984,
0.02919781394302845,
0.027890779078006744,
-0.002277705119922757,
0.005881260149180889,
-0.049884721636772156,
-0.04685933515429497,
0.017291339114308357,
-0.07725325226783752,
-0.004005295690149069,
-0.07872898876667023,
-0.07629788666963577,
-0.035515304654836655,
-0.07369444519281387,
-0.046943649649620056,
-0.008588531985878944,
3.117224775994498e-33,
-0.06376376748085022,
0.07068392634391785,
-0.09900470077991486,
-0.04142474755644798,
-0.05417294800281525,
-0.021611616015434265,
0.047426484525203705,
0.006639709230512381,
0.06628664582967758,
0.02731831930577755,
0.005111610051244497,
0.03495554253458977,
0.01336253248155117,
-0.015671228989958763,
-0.0629960149526596,
-0.03361336141824722,
-0.057769495993852615,
-0.04075955972075462,
0.0057192896492779255,
-0.03095506876707077,
-0.03151414915919304,
0.028117071837186813,
0.010128800757229328,
0.004339145962148905,
0.018112925812602043,
0.05575426295399666,
0.06719312071800232,
-0.03705766797065735,
0.019272135570645332,
0.09479346871376038,
-0.0038623809814453125,
-0.06944166868925095,
0.04814118891954422,
0.028226075693964958,
0.009504192508757114,
-0.0846368819475174,
0.026623813435435295,
0.011749381199479103,
0.06415579468011856,
-0.018739013001322746,
0.08205463737249374,
0.03573252260684967,
0.055349692702293396,
-0.030733028426766396,
0.0030227811075747013,
-0.021069727838039398,
0.018833911046385765,
0.0839274674654007,
-0.04346991702914238,
-0.04993315041065216,
-0.04088948667049408,
-0.06612271070480347,
-0.05405239015817642,
-0.013708867132663727,
0.04450991377234459,
0.02571759931743145,
0.03404468670487404,
-0.03589456155896187,
0.0720132440328598,
-0.015372741967439651,
-0.013447820208966732,
-0.10117825865745544,
0.08097701519727707,
0.009273574687540531,
-0.029240958392620087,
0.004674158990383148,
0.01426104735583067,
-0.023023024201393127,
-0.013649014756083488,
0.04421999678015709,
0.03787430748343468,
0.03689057007431984,
-0.11434118449687958,
-0.048899758607149124,
-0.06006154045462608,
0.047949329018592834,
-0.011136223562061787,
0.06939148157835007,
-0.011418947018682957,
-0.0837293267250061,
-0.03982200846076012,
0.012051790952682495,
0.04016907140612602,
-0.03214588761329651,
-0.09053593873977661,
0.004030831623822451,
0.13588710129261017,
0.06065797805786133,
0.008787855505943298,
-0.017792444676160812,
0.0015248669078573585,
0.04013393819332123,
-0.010127768851816654,
0.006726089399307966,
0.08717994391918182,
7.580109493593875e-32,
0.014222467318177223,
0.06863290071487427,
-0.05034295469522476,
-0.10769671201705933,
0.025501547381281853,
-0.007771725300699472,
-0.046187546104192734,
0.04391134902834892,
0.059887826442718506,
-0.0745292529463768,
-0.009308154694736004,
0.04790113866329193,
-0.06216663494706154,
0.058498185127973557,
-0.09123285114765167,
-0.033652305603027344,
-0.02854161337018013,
0.020794128999114037,
-0.0020471513271331787,
-0.033469706773757935,
-0.02436743676662445,
0.03199979290366173,
-0.044979725033044815,
0.07841044664382935,
0.03985735401511192,
0.06756926327943802,
-0.14928215742111206,
0.045914266258478165,
-0.023570355027914047,
-0.028824735432863235,
0.060037292540073395,
0.004622276872396469,
0.013469932600855827,
-0.017681287601590157,
0.039303410798311234,
-0.06394781917333603,
0.03786314278841019,
0.012257690541446209,
0.019663555547595024,
0.0305195190012455,
-0.04360850527882576,
-0.06903760135173798,
-0.020996930077672005,
-0.0013747893972322345,
0.027974797412753105,
-0.026206519454717636,
-0.0220070481300354,
-0.0010386627400293946,
-0.01138413418084383,
-0.0172492153942585,
-0.04726354777812958,
-0.028642671182751656,
-0.0068350983783602715,
0.012442619539797306,
-0.029059985652565956,
-0.01669253036379814,
-0.026816142722964287,
0.026210615411400795,
0.011342817917466164,
-0.012693420983850956,
-0.09320325404405594,
0.07189703732728958,
-0.08098262548446655,
0.06493941694498062
] |
18
Quadratics
After completing this chapter you should be able to:
β Solve quadratic equations using fact
orisation, the quadratic
formula and completing the square β pages 19 β 24
β Read and use f(x) notation when working with
functions β pages 25 β 27
β Sketch the graph and find the turning point of a quadratic function
β pages 27 β 30
β Find and interpret the discriminant o f a quadratic
expression β pages 30 β 32
β Use and apply models that involv e quadratic
functions β pages 32 β 35Objectives
1 Solve the following equations:
a 3x
+ 6 = x β 4
b 5(x
+ 3) = 6(2x β 1)
c 4x2 = 100
d (x β
8)2 = 64 β GCSE Mathematics
2 Factorise the following expressions:
a x2 + 8x + 15 b x2 + 3x β 10
c 3x2 β 14x β 5 d x2 β 400
β Section 1.3
3 Sketch the graphs of the following equations, labelling the points wher
e each
graph crosses the axes:
a y =
3x β 6 b y =
10 β 2x
c x +
2y = 18 d y =
x2
β GCSE Mathematics
4 Solve the following inequalities:
a x +
8 , 11 b 2x
β 5 > 13
c 4x
β 7 < 2 (x β 1) d 4 β
x , 11
β GCSE MathematicsPrior knowledge check
Quadratic functions are used to model projectile motion. Whenever an object is thrown or launched, its path will approximately follow the shape of a parabola.
β Mixed exercise Q112
|
[
0.016577165573835373,
0.06907497346401215,
0.011500346474349499,
-0.04895270988345146,
0.05113945156335831,
0.09139422327280045,
-0.08206266164779663,
0.0018851568456739187,
-0.13797101378440857,
0.05486544221639633,
0.01460819598287344,
-0.08882471174001694,
0.008212676271796227,
0.025523604825139046,
0.010412650182843208,
0.04081058129668236,
-0.09664537757635117,
0.05729135125875473,
-0.034762002527713776,
-0.06588911265134811,
0.0689634457230568,
-0.024425150826573372,
-0.02895229682326317,
-0.030876316130161285,
0.059912458062171936,
-0.042389076203107834,
-0.0021412603091448545,
-0.029158594086766243,
-0.04256780445575714,
-0.04208696261048317,
-0.032875701785087585,
0.05173429474234581,
0.06928732991218567,
-0.031085936352610588,
-0.06044590100646019,
-0.007891282439231873,
0.10392449051141739,
-0.016333557665348053,
-0.021523116156458855,
-0.03814634308218956,
-0.0147068677470088,
0.06101235747337341,
-0.06314923614263535,
0.03631763905286789,
0.09608788043260574,
-0.07815946638584137,
-0.013389116153120995,
0.008867597207427025,
0.04580206051468849,
-0.11338987946510315,
-0.01493642758578062,
-0.019024919718503952,
-0.01821925677359104,
0.0717673972249031,
0.014965449459850788,
-0.06468527019023895,
-0.05500775948166847,
0.09092532098293304,
-0.007365584373474121,
-0.0011761285131797194,
-0.01862078346312046,
-0.008862639777362347,
0.023217296227812767,
0.0453956164419651,
0.03925560414791107,
0.06692765653133392,
0.06259245425462723,
-0.04530148208141327,
0.013677510432898998,
0.1239493116736412,
-0.06345512717962265,
0.01498682051897049,
-0.08969703316688538,
0.04422716796398163,
0.051572639495134354,
-0.016381995752453804,
-0.021072393283247948,
-0.024330835789442062,
0.0007526194676756859,
-0.03773054853081703,
-0.03227577358484268,
0.0033061911817640066,
0.044978801161050797,
0.07159869372844696,
0.016967615112662315,
-0.043562114238739014,
0.04121297225356102,
0.01871822215616703,
-0.017087753862142563,
-0.02387206070125103,
0.03218819573521614,
-0.057992637157440186,
0.002821422414854169,
-0.08036324381828308,
-0.03354182466864586,
-0.036791104823350906,
-0.043752942234277725,
-0.06389250606298447,
0.08888348191976547,
0.031506940722465515,
0.021223563700914383,
-0.05175379663705826,
0.04664360731840134,
0.00757836876437068,
-0.11675623059272766,
-0.014580434188246727,
0.011492479592561722,
-0.02467123232781887,
0.04941737651824951,
-0.041830290108919144,
-0.127852663397789,
0.005665639415383339,
-0.0024002392310649157,
0.0040155746974051,
0.08351918309926987,
-0.043348122388124466,
0.07088892161846161,
-0.0783257931470871,
-0.024364646524190903,
-0.00833186786621809,
-0.03862449526786804,
-0.02070154808461666,
0.09967893362045288,
-0.032135553658008575,
0.02425062656402588,
0.039985544979572296,
-0.020542427897453308,
0.03804335370659828,
-0.03135877475142479,
-0.006406824104487896,
0.04192252457141876,
0.02159852907061577,
-0.056274790316820145,
-0.05285187065601349,
0.017143120989203453,
-0.0256411861628294,
-0.07731872797012329,
0.059236813336610794,
-0.04201348498463631,
-0.02330208756029606,
-0.00019692396745085716,
0.12084263563156128,
0.06033164635300636,
0.01766824536025524,
-0.00012886314652860165,
-0.04289311543107033,
-0.03465648368000984,
0.03754948079586029,
-0.03285559266805649,
-0.01828651875257492,
0.0006079273298382759,
-0.03122406080365181,
0.009816550649702549,
0.07664533704519272,
-0.007253935094922781,
-0.08875739574432373,
0.030189912766218185,
-0.009045900776982307,
0.009877867996692657,
0.05191183090209961,
-0.033432066440582275,
-0.0018145196372643113,
-0.03739858791232109,
0.06766393035650253,
0.035771556198596954,
-0.0061318050138652325,
0.02558049187064171,
-0.0213867649435997,
-0.07443926483392715,
0.06391657888889313,
0.008202288299798965,
-0.00852323416620493,
0.004257692024111748,
0.0881420448422432,
0.022344740107655525,
-0.03587162122130394,
0.1508079469203949,
0.005463337991386652,
0.0015741089591756463,
0.07845447957515717,
0.035404272377491,
-0.010748712345957756,
-0.034322284162044525,
0.000657429511193186,
0.02643614634871483,
0.03589192032814026,
0.023458553478121758,
0.030113358050584793,
0.0087353540584445,
-0.05284465104341507,
0.0039923046715557575,
-0.07006946206092834,
-0.015578089281916618,
0.015657050535082817,
-0.08880414068698883,
-0.047637809067964554,
-0.018887920305132866,
0.010381564497947693,
-0.10654223710298538,
0.011427091434597969,
0.03323178365826607,
-0.07968504726886749,
0.043204665184020996,
0.023083128035068512,
0.0029589631594717503,
-0.052633874118328094,
-0.11730269342660904,
-0.01111135445535183,
0.06408273428678513,
0.10300160944461823,
-0.0835014060139656,
0.007377327419817448,
0.03184296563267708,
-0.062296994030475616,
-0.004273534752428532,
-0.025519300252199173,
0.017854955047369003,
-0.08427717536687851,
-0.13643397390842438,
0.033742718398571014,
-0.07192615419626236,
-0.002086078515276313,
0.05487693101167679,
-4.772615430223954e-33,
-0.018641823902726173,
0.006549261510372162,
-0.11972284317016602,
-0.030316505581140518,
0.013073991984128952,
-0.03304409235715866,
0.01885080151259899,
-0.1203572154045105,
0.09859704226255417,
-0.017922595143318176,
0.021396860480308533,
0.00433385232463479,
-0.021086333319544792,
0.044313665479421616,
-0.06682579964399338,
-0.0325738787651062,
0.013466431759297848,
-0.02442634105682373,
0.02071721851825714,
-0.11859089881181717,
0.007731970865279436,
0.07277733087539673,
-0.013692736625671387,
-0.028057843446731567,
-0.03184698894619942,
0.08482132107019424,
0.04945332556962967,
-0.04501248896121979,
0.01708081364631653,
0.06555123627185822,
-0.024801988154649734,
0.03994501382112503,
0.05735388770699501,
-0.00017031552852131426,
-0.03618590533733368,
-0.04454252868890762,
0.021717306226491928,
-0.03123575635254383,
0.031094679608941078,
0.03548593446612358,
0.1447768360376358,
-0.019261183217167854,
0.07776182889938354,
-0.0329756923019886,
0.006405004300177097,
0.04033440724015236,
0.06966715306043625,
-0.0042367959395051,
-0.03230142220854759,
0.01592506468296051,
-0.0027116185519844294,
-0.0022170234005898237,
-0.0257083959877491,
0.03329683467745781,
-0.002537292428314686,
0.010375600308179855,
0.02512778341770172,
-0.04274903982877731,
0.09676504880189896,
0.02197747677564621,
-0.022004419937729836,
0.0021936811972409487,
-0.010529511608183384,
0.06123514100909233,
-0.07037538290023804,
-0.06179945543408394,
-0.0067336903885006905,
0.000629144546110183,
-0.028160033747553825,
0.0289912186563015,
-0.021858736872673035,
0.0046109361574053764,
-0.031602729111909866,
-0.07364573329687119,
-0.04180672764778137,
-0.045122213661670685,
-0.021957658231258392,
0.03389928489923477,
0.027909740805625916,
-0.03670518472790718,
-0.09993938356637955,
0.01389659009873867,
-0.011655379086732864,
-0.04017215222120285,
-0.09774550795555115,
-0.006688275840133429,
0.09258157014846802,
0.028701862320303917,
0.044949863106012344,
-0.03266990929841995,
0.020144738256931305,
-0.007481846492737532,
0.03733920678496361,
-0.03227267786860466,
0.0773683711886406,
6.944208779848472e-32,
-0.04112771898508072,
0.06699132919311523,
-0.01755291409790516,
0.04161573946475983,
-0.016496814787387848,
-0.03356313332915306,
-0.0654950737953186,
0.04850080981850624,
0.03091723471879959,
-0.02696964144706726,
0.11684663593769073,
-0.02566308155655861,
-0.035428136587142944,
-0.0003983824863098562,
-0.09065693616867065,
-0.05251336470246315,
-0.003479256061837077,
0.11731131374835968,
0.0014196464326232672,
-0.029329808428883553,
-0.01061042957007885,
0.06056113913655281,
0.015578712336719036,
0.03177063167095184,
0.005399289540946484,
0.07776402682065964,
-0.08065547049045563,
-0.05248725786805153,
0.009106521494686604,
0.034738823771476746,
0.08929671347141266,
0.07997741550207138,
-0.01932976022362709,
-0.04166131466627121,
0.0395156666636467,
-0.044151704758405685,
0.05953797325491905,
-0.05640142410993576,
0.025350326672196388,
-0.0160212479531765,
-0.01008518785238266,
-0.08130701631307602,
-0.08638910949230194,
0.021931461989879608,
0.0029692708048969507,
-0.03351012244820595,
-0.049286138266325,
-0.07973048090934753,
0.04043367877602577,
0.04970601573586464,
-0.01256448682397604,
0.03643083572387695,
0.051201507449150085,
-0.025610141456127167,
0.02698969654738903,
-0.011154730804264545,
-0.05949590355157852,
0.02180970273911953,
-0.03439737856388092,
-0.026921140030026436,
0.031789932399988174,
0.11825385689735413,
-0.10217120498418808,
0.05034361034631729
] |
19Quadratics
2.1 Solving quadratic equations
A quadratic equation can be written in the form ax2 + bx + c = 0, where a, b and c are real constants,
and a β 0. Quadratic equations can have one, two, or no real solutions.
β To solve a quadratic equation by factorising:
β’ Writ
e the equation in the form ax2 + bx + c = 0
β’ Factorise the left-hand side
β’ Set each factor equal to zero and solve to find the value(s) of x
Example 1 The s olutions to an
equation are sometimes called
the roots of the equation.Notation
The symbol β means βimplies thatβ .
This statement says βIf x + 3 = 0, then x = β3β.Notation
a x2 β 2x β 15 = 0
(x
+ 3)( x β 5) = 0
The
n either x + 3 = 0 β
x = β 3
or x β
5 = 0 β
x = 5
So x = β 3 and x = 5 are the two solutions
of the equation.
b x2 = 9x
x2 β 9 x = 0
x(x β
9) = 0
Th
en either x = 0
or x β
9 = 0 β
x = 9
The solutions are x = 0 and x = 9.
c 6x2 + 13 x β 5 = 0
(3x β
1)(2x + 5) = 0
The
n either 3 x β 1 = 0 β
x = 1 __ 3
or 2x
+ 5 = 0 β
x = β 5 __ 2
The s
olutions are x = 1 __ 3 and x = β 5 __ 2
d x2 β 5 x + 18 = 2 + 3 x
x2 β 8 x + 16 = 0
(x β
4)(x β 4) = 0
Th
en either x β 4 = 0 β
x = 4
or x β
4 = 0 β
x = 4
β x = 4Factorise. The s igns of the solutions are
opposite to the signs of the constant terms in
each factor.Watch outSolve the following equations:
a x2 β 2x β 15 = 0 b x2 = 9x
c 6x2 + 13x β 5 = 0 d x2 β 5x + 18 = 2 + 3xIf the product of the factors is zero, one of the
factors must be zero.Factorise the quadratic. β Section 1.3
A quadratic equation with two distinct factors has two distinct solutions.
Be careful not to divide both sides by x, since x may have the value 0. Instead, rearrange into the form ax
2 + bx + c = 0.
Factorise.
Solutions to quadratic equations do not have to be integers.
The quadratic equation (px + q)(rx + s) = 0 will have solutions x = β
q __ p and x = β s __ r .
When a quadratic equation has
e
xactly one root it is called a repeated root. You
can also say that the equation has two equal roots.NotationRearrange into the form ax2 + bx + c = 0.
Factorise.
|
[
-0.00751400925219059,
0.10765448957681656,
0.03081025928258896,
0.036916669458150864,
-0.007839293219149113,
0.0071492320857942104,
-0.03982795774936676,
-0.012176801450550556,
-0.051707908511161804,
0.07581004500389099,
-0.01880301721394062,
-0.07049570232629776,
0.03265965357422829,
-0.0009297432843595743,
0.08823050558567047,
0.03179856017231941,
-0.1035531610250473,
0.014980899170041084,
-0.023358307778835297,
-0.017254386097192764,
0.02749559096992016,
-0.1297367364168167,
-0.026151422411203384,
0.010936061851680279,
0.00592498853802681,
-0.05435078218579292,
-0.013968011364340782,
-0.032621175050735474,
0.01145000196993351,
0.027066804468631744,
-0.05326886102557182,
0.08236739039421082,
0.03746619448065758,
0.020899055525660515,
0.0278758704662323,
0.038681164383888245,
0.06740404665470123,
0.03404756262898445,
-0.052305612713098526,
-0.03711554780602455,
-0.037898022681474686,
0.08274078369140625,
-0.0366695411503315,
0.0039008809253573418,
0.037605948746204376,
-0.028059421107172966,
-0.025118254125118256,
-0.08613356202840805,
0.09466569125652313,
-0.04659311845898628,
0.004201111849397421,
0.00643874891102314,
-0.0814579650759697,
0.026909463107585907,
-0.07004179060459137,
-0.011368864215910435,
-0.0574786514043808,
0.022091353312134743,
-0.035670265555381775,
0.030907152220606804,
0.004568654112517834,
-0.03860609978437424,
0.14683426916599274,
0.051210250705480576,
0.07248745858669281,
0.02923751063644886,
-0.03303900733590126,
0.003598600160330534,
0.015041396021842957,
0.032590366899967194,
0.005665637087076902,
-0.0008731876732781529,
0.010334379971027374,
0.0480530746281147,
0.055230725556612015,
-0.03498569875955582,
-0.049506835639476776,
0.03326765075325966,
0.0983881950378418,
0.05529170483350754,
0.010296930558979511,
-0.05058981850743294,
0.039533112198114395,
0.014343537390232086,
-0.003188494825735688,
-0.04877880960702896,
0.14663352072238922,
0.04988260567188263,
0.040943045169115067,
0.019226009026169777,
0.010070701129734516,
-0.04721628502011299,
0.023580675944685936,
-0.06784715503454208,
-0.022110862657427788,
-0.1038365364074707,
0.03712780401110649,
0.07655417174100876,
-0.009917703457176685,
-0.06420113891363144,
0.041700851172208786,
-0.06376568228006363,
-0.0048853433690965176,
0.0308840349316597,
-0.05760226398706436,
-0.02825140580534935,
-0.06800279766321182,
0.004119290504604578,
0.0885901004076004,
-0.034275591373443604,
-0.11157949268817902,
-0.022951947525143623,
0.008698083460330963,
0.04858803376555443,
0.02613893710076809,
-0.08281673491001129,
0.10445677489042282,
-0.014619638212025166,
0.032542333006858826,
-0.0847921371459961,
-0.04086761176586151,
-0.06514554470777512,
0.04226793348789215,
-0.07644153386354446,
0.05286315456032753,
0.055046722292900085,
0.04293229058384895,
0.022072644904255867,
-0.07480937987565994,
-0.05795453488826752,
0.011312159709632397,
-0.03297377750277519,
-0.1083880364894867,
-0.06166738644242287,
0.06495234370231628,
-0.0437227301299572,
-0.044029075652360916,
0.12200256437063217,
-0.010063501074910164,
-0.07847265899181366,
0.043744493275880814,
0.06890374422073364,
-0.003423871472477913,
0.050131622701883316,
0.044341035187244415,
-0.06044134125113487,
-0.007640977390110493,
-0.0438544861972332,
-0.03574703261256218,
-0.02081884630024433,
-0.060568053275346756,
0.006590624339878559,
0.005219275131821632,
0.04599453881382942,
0.03186998516321182,
-0.1215514987707138,
-0.03845177963376045,
-0.08412358909845352,
-0.054663922637701035,
0.07707827538251877,
-0.052260302007198334,
-0.03940870612859726,
-0.07237163186073303,
0.02697722800076008,
-0.08390364050865173,
0.004630410578101873,
-0.0045156278647482395,
0.035665690898895264,
0.002229722449555993,
0.05047757551074028,
-0.026346804574131966,
0.0020944077987223864,
0.03020331636071205,
0.05812545493245125,
-0.01589847542345524,
0.009877204895019531,
0.004379570484161377,
0.026887275278568268,
0.06655113399028778,
0.06547427922487259,
0.041750941425561905,
0.05332905799150467,
-0.06848355382680893,
0.046974536031484604,
0.03728358447551727,
0.006447015795856714,
-0.011646603234112263,
-0.10912510007619858,
-0.024645881727337837,
-0.079787977039814,
0.020590730011463165,
-0.038919925689697266,
-0.021302398294210434,
0.07263916730880737,
-0.0663783848285675,
-0.020117882639169693,
-0.006117168348282576,
-0.033977266401052475,
-0.03843195363879204,
0.03694802522659302,
0.033415548503398895,
-0.07359930127859116,
0.04180171340703964,
0.018864251673221588,
-0.016961367800831795,
0.055857885628938675,
-0.1019602045416832,
-0.011456972919404507,
-0.02066916413605213,
0.09206251800060272,
-0.04086678475141525,
-0.017022117972373962,
0.05015689134597778,
-0.0488763265311718,
-0.032260145992040634,
-0.0619981624186039,
-0.05417337641119957,
-0.007473889738321304,
-0.030904067680239677,
0.059966664761304855,
-0.07987014204263687,
0.013134080916643143,
-0.003212233539670706,
-9.703582439131147e-33,
0.005064823664724827,
-0.029364561662077904,
-0.12917321920394897,
0.014336295425891876,
-0.062351055443286896,
0.06251317262649536,
0.054853253066539764,
-0.12983626127243042,
0.14401333034038544,
-0.04044564440846443,
0.055309150367975235,
0.01079499814659357,
-0.06859949976205826,
-0.00025091524003073573,
-0.07036614418029785,
0.02701478637754917,
-0.052884310483932495,
0.038716379553079605,
0.02670012228190899,
-0.04871653392910957,
0.07475263625383377,
0.01816791296005249,
0.0035817231982946396,
-0.03027452901005745,
0.027578961104154587,
0.002421530894935131,
0.035594116896390915,
0.006154782138764858,
0.05746837705373764,
0.012722858227789402,
-0.053195755928754807,
0.005902804434299469,
0.05997374653816223,
0.01531416829675436,
-0.012010552920401096,
-0.02741738222539425,
0.004080865532159805,
0.01086745597422123,
-0.006866420153528452,
-0.04107644781470299,
-0.018711240962147713,
-0.004443932790309191,
0.09846142679452896,
0.04155489057302475,
0.02672324888408184,
0.041384369134902954,
0.0144991185516119,
0.03648391366004944,
0.01158865261822939,
0.021453112363815308,
-0.040408335626125336,
-0.06445655226707458,
-0.016294820234179497,
0.008282283321022987,
0.031355731189250946,
0.08803774416446686,
0.013470477424561977,
-0.061823613941669464,
-0.0406680628657341,
-0.036103811115026474,
0.005713994614779949,
-0.042711518704891205,
0.03493592515587807,
0.046779949218034744,
0.016971437260508537,
0.06008675694465637,
-0.014772157184779644,
0.0055925725027918816,
-0.027057604864239693,
0.0031520389020442963,
-0.0050312490202486515,
0.030894942581653595,
-0.059730857610702515,
0.006917144171893597,
-0.04444582015275955,
0.057040441781282425,
-0.12168103456497192,
0.045493483543395996,
0.029779288917779922,
-0.05073846876621246,
0.03517552837729454,
0.06275302916765213,
-0.006156265735626221,
-0.05800704285502434,
-0.05484117567539215,
-0.03151674568653107,
0.06340665370225906,
0.06596314907073975,
0.008694457821547985,
-0.03322742506861687,
0.054087888449430466,
-0.061824824661016464,
-0.021486327052116394,
-0.008050513453781605,
0.10175352543592453,
6.410831156898736e-32,
-0.029499437659978867,
-0.008376380428671837,
-0.019764786586165428,
0.025372281670570374,
0.03439958393573761,
0.025220073759555817,
-0.03556625545024872,
-0.002459562150761485,
0.019843004643917084,
-0.06267610192298889,
0.0036865021102130413,
-0.005805158521980047,
-0.010229661129415035,
0.015263169072568417,
-0.05766092613339424,
0.023096822202205658,
-0.004369029775261879,
0.07136505842208862,
-0.02270175702869892,
0.025149859488010406,
-0.04284650832414627,
0.031229956075549126,
-0.05597769841551781,
0.03821130469441414,
0.023836452513933182,
0.05056116357445717,
-0.015135814435780048,
-0.013443835079669952,
0.012302654795348644,
-0.03197113052010536,
0.08682586252689362,
0.06766275316476822,
-0.015372182242572308,
0.024071644991636276,
0.05288318172097206,
0.021040501073002815,
0.11513518542051315,
0.012754182331264019,
-0.015927629545331,
0.00424569146707654,
0.023296086117625237,
-0.05025860294699669,
-0.09131138771772385,
-0.018823478370904922,
-0.0963391438126564,
-0.07849148660898209,
-0.032948024570941925,
-0.13437220454216003,
0.06980817019939423,
0.01129255909472704,
0.03205018863081932,
-0.01944964937865734,
0.054774727672338486,
0.0582142174243927,
0.014207867905497551,
-0.11218681931495667,
-0.07188789546489716,
-0.02638905867934227,
0.030360111966729164,
-0.06189969927072525,
-0.07914307713508606,
0.06625057011842728,
-0.038328301161527634,
-0.0606926828622818
] |
20
Chapter 2
In some cases it may be more straightforward to solve a quadratic equation without factorising.
Example 2
Solve the following equations
a (2x
β 3)2 = 25 b (x β 3)2 = 7
a (2x β 3)2 = 25
2x
β 3 = Β±5
2x = 3 Β±
5
The
n either 2x = 3 + 5 β
x = 4
or 2x = 3 β
5 β x = β 1
The solutions are x = 4 and x = β 1
b (x
β 3)2 = 7
x β
3 = Β± ββ―__
7
x = 3 Β±
ββ―__
7
The s
olutions are x = 3 + ββ―__
7 and
x = 3 β ββ―__
7 The symbol Β± lets you write two
statements in one line of working. You say
βplus or minusβ.Notation
Add 3 to both sides.
1 Solve the following equations using factorisation:
a x2 + 3x + 2 = 0 b x2 + 5x + 4 = 0 c x2 + 7x + 10 = 0 d x2 β x β 6 = 0
e x2 β 8x + 15 = 0 f x2 β 9x + 20 = 0 g x2 β 5x β 6 = 0 h x2 β 4x β 12 = 0
2 Solve the follo
wing equations using factorisation:
a x2 = 4x b x2 = 25x c 3x2 = 6x d 5x2 = 30x
e 2x2 + 7x + 3 = 0 f 6x2 β 7x β 3 = 0 g 6x2 β 5x β 6 = 0 h 4x2 β 16x + 15 = 0
3 Solve the follo
wing equations:
a 3x2 + 5x = 2 b (2x β 3)2 = 9 c (x β 7)2 = 36 d 2x2 = 8 e 3x2 = 5
f (x
β 3)2 = 13 g (3x β 1)2 = 11 h 5x2 β 10x2 = β7 + x + x2
i 6x2 β 7 = 11x j 4x2 + 17x = 6x β 2x2
4 This shape has an area of 44 m2.
Find the value of x.
5 Solve the equation 5
x + 3 = ββ―______ 3x + 7 .2x mx m
x m(x + 3) mP
PExercise 2A
Divide the shape into two sections:Problem-solvingTake the square root of both sides.
Remember 52 = (β5)2 = 25.
Take square roots of both sides.
You can leave your answer in surd form.
|
[
-0.04855820909142494,
0.10772905498743057,
0.05654939264059067,
-0.012505405582487583,
0.01890873908996582,
0.038533926010131836,
-0.024093758314847946,
0.012748409993946552,
-0.08445215225219727,
0.0044477516785264015,
0.0473746582865715,
-0.0657033696770668,
0.033453140407800674,
0.022697998210787773,
0.023176079615950584,
0.011865661479532719,
-0.060559507459402084,
0.010782988741993904,
-0.052084896713495255,
-0.04896289110183716,
-0.002340950071811676,
-0.03228718042373657,
-0.007124040741473436,
0.0050192526541650295,
0.0693139135837555,
-0.08424457162618637,
-0.025490840896964073,
-0.06834905594587326,
-0.0047498587518930435,
-0.0035245923791080713,
-0.03313668444752693,
0.07765275985002518,
0.014431264251470566,
-0.007191590964794159,
0.025193581357598305,
0.04959184303879738,
0.07321643829345703,
0.05381646752357483,
-0.08040113002061844,
-0.10076938569545746,
0.021315088495612144,
0.03428443521261215,
-0.07374720275402069,
0.027266530320048332,
0.05755392462015152,
-0.07655074447393417,
-0.025598803535103798,
-0.056153807789087296,
0.02400553598999977,
-0.08148375898599625,
-0.030868908390402794,
0.05605563148856163,
-0.02918919362127781,
0.010453391820192337,
-0.06378445029258728,
0.02912275865674019,
-0.025728125125169754,
0.061294641345739365,
-0.010807536542415619,
0.028563521802425385,
0.016638072207570076,
0.015715032815933228,
0.050162144005298615,
0.0809832215309143,
0.05512065812945366,
0.000684386002831161,
-0.02256869152188301,
-0.033973850309848785,
0.03450585901737213,
0.05363902077078819,
0.0052081141620874405,
0.03346347063779831,
-0.033610280603170395,
0.059812553226947784,
0.02216421067714691,
-0.0271740909665823,
-0.08210203051567078,
-0.0019273916259407997,
0.01375430915504694,
-0.017515206709504128,
0.012970499694347382,
-0.011969402432441711,
0.03730666637420654,
-0.0077958363108336926,
-0.011414755135774612,
-0.10154574364423752,
0.07619060575962067,
0.036317311227321625,
0.04346688464283943,
0.0007324177422560751,
-0.010051272809505463,
-0.06154351681470871,
0.02142319083213806,
-0.08750027418136597,
0.013186234049499035,
-0.1001356989145279,
0.02792772464454174,
0.0390310175716877,
-0.008040975779294968,
-0.06585650891065598,
0.031223343685269356,
-0.02554808184504509,
0.005863157566636801,
0.006928710732609034,
-0.07524339109659195,
-0.013274861499667168,
-0.08144145458936691,
-0.021452082321047783,
0.10803617537021637,
-0.059525955468416214,
-0.1416628658771515,
-0.04023396968841553,
0.048562537878751755,
0.0046360082924366,
0.0369284451007843,
-0.11239595711231232,
0.11265601962804794,
-0.07155516743659973,
0.04757319763302803,
-0.03506670892238617,
0.014403794892132282,
-0.07440735399723053,
0.07789807766675949,
-0.06746572256088257,
0.08997055888175964,
0.07048210501670837,
0.011830316856503487,
-0.01816430129110813,
-0.05231406167149544,
-0.0816965103149414,
0.027242936193943024,
-0.028612572699785233,
-0.09111034870147705,
-0.061208516359329224,
0.014949564822018147,
-0.014574218541383743,
0.009912330657243729,
0.0937265083193779,
0.018241532146930695,
0.012802032753825188,
0.018510261550545692,
0.07405240088701248,
0.08258848637342453,
0.09939545392990112,
-0.029215367510914803,
-0.07810188829898834,
-0.015941260382533073,
0.023511717095971107,
0.0028234696947038174,
-0.02544727362692356,
-0.056485120207071304,
-0.03524326905608177,
-0.06964551657438278,
0.01752415858209133,
0.07507702708244324,
-0.10385279357433319,
0.04164037108421326,
-0.02379631996154785,
-0.02959306910634041,
0.024237753823399544,
-0.09780838340520859,
-0.03838464617729187,
-0.0754619836807251,
0.018556764349341393,
-0.028829528018832207,
0.022676458582282066,
-0.09408726543188095,
0.0075296140275895596,
0.0168503038585186,
0.02505960687994957,
-0.028677107766270638,
0.04760781675577164,
0.02755192667245865,
0.06978612393140793,
-0.0016893213614821434,
0.0411435104906559,
-0.005341361742466688,
0.004313056357204914,
0.05015898868441582,
0.11620038747787476,
0.11317399144172668,
-0.018610844388604164,
0.033961378037929535,
0.036838747560977936,
0.02022458240389824,
-0.010168921202421188,
0.0307169146835804,
-0.007143862545490265,
-0.00943872332572937,
-0.11706089228391647,
-0.033021677285432816,
-0.041230615228414536,
0.007197091821581125,
0.01843566633760929,
-0.05293304845690727,
-0.018163586035370827,
-0.029277663677930832,
-0.06508345156908035,
-0.1092490628361702,
0.057562436908483505,
0.06529191136360168,
-0.05656878277659416,
0.11682137846946716,
-0.0744287371635437,
0.035288359969854355,
0.03781229257583618,
-0.06998217850923538,
-0.028727686032652855,
0.05670102685689926,
0.0976172387599945,
-0.040613818913698196,
0.0732090100646019,
0.002971792593598366,
-0.02849041484296322,
-0.005974673666059971,
-0.04826780781149864,
-0.002776363166049123,
-0.01888197474181652,
-0.012496892362833023,
0.01931493729352951,
-0.06552965939044952,
0.03641202673316002,
0.050774939358234406,
-1.3434954156946334e-32,
-0.05047426372766495,
-0.10342749208211899,
-0.12123540788888931,
0.01965213753283024,
-0.03175562247633934,
-0.003256395924836397,
0.026731284335255623,
-0.1802964061498642,
0.08509166538715363,
-0.018780266866087914,
-0.0309661366045475,
0.011345181614160538,
-0.0047514052130281925,
0.03616366535425186,
-0.02057848498225212,
0.011216298677027225,
-0.07523319870233536,
0.02758215367794037,
-0.007743239402770996,
-0.06446170061826706,
0.030766911804676056,
-0.003200035309419036,
-0.0036927922628819942,
-0.08655202388763428,
0.0008429296431131661,
0.051259905099868774,
0.042819395661354065,
-0.012131071649491787,
0.034868862479925156,
0.07947061210870743,
-0.003918347880244255,
-0.0049877772107720375,
0.06322227418422699,
-0.017816023901104927,
-0.02514888346195221,
-0.033479493111371994,
-0.028842205181717873,
0.025449851527810097,
-0.031084975227713585,
-0.06070834398269653,
0.062411874532699585,
-0.0015317662619054317,
0.09136325120925903,
-0.04457199573516846,
-0.013664989732205868,
0.022220782935619354,
0.016786696389317513,
0.024547763168811798,
-0.058562882244586945,
0.03892785683274269,
-0.02578500285744667,
0.03561955690383911,
-0.014889317564666271,
0.07553398609161377,
-0.04599480330944061,
0.006933803670108318,
0.052008502185344696,
0.0072862738743424416,
-0.05869603529572487,
-0.04004606977105141,
-0.015016899444162846,
-0.04273609071969986,
0.02785346284508705,
0.03170868754386902,
-0.0422038771212101,
0.06170734763145447,
0.0710361897945404,
0.06758227199316025,
-0.002479103859513998,
0.06564222276210785,
-0.06424818933010101,
0.03868584334850311,
-0.027030376717448235,
0.048529673367738724,
0.010500547476112843,
0.08142843097448349,
-0.0924566388130188,
0.046284373849630356,
-0.03725215420126915,
-0.04210492968559265,
0.009728892706334591,
0.023869972676038742,
-0.024720801040530205,
-0.07709576934576035,
-0.05680396035313606,
-0.027846839278936386,
0.05367071181535721,
-0.02621125988662243,
0.003729806514456868,
-0.09526016563177109,
0.021000631153583527,
-0.08561644703149796,
0.049209803342819214,
-0.0014668453950434923,
0.09804555773735046,
1.0039665677986046e-31,
-0.00864747166633606,
0.0171111561357975,
-0.025419628247618675,
-0.012936661951243877,
-0.0018315943889319897,
-0.0194639191031456,
0.0021507784258574247,
0.05202353373169899,
0.02894602157175541,
-0.022780412808060646,
-0.01990489289164543,
0.052641406655311584,
-0.043760571628808975,
-0.005308210384100676,
-0.06664277613162994,
-0.008984253741800785,
-0.002706094877794385,
0.0479099415242672,
-0.03756454959511757,
0.03919188305735588,
-0.017851106822490692,
0.027151798829436302,
-0.08847146481275558,
-0.023371299728751183,
-0.00322935963049531,
0.058927983045578,
0.0016674797516316175,
-0.0048590232618153095,
0.02895057387650013,
-0.012942169792950153,
0.014851781539618969,
0.054599739611148834,
-0.0008902594563551247,
-0.03930245712399483,
0.04840632528066635,
0.059369612485170364,
0.11878068000078201,
0.05912748724222183,
-0.04285300895571709,
-0.047135524451732635,
-0.007036859635263681,
-0.04944644868373871,
-0.052304599434137344,
-0.010914224199950695,
-0.03628907352685928,
-0.06477116793394089,
-0.06411180645227432,
-0.09629389643669128,
0.029533807188272476,
0.05517147108912468,
0.03868720307946205,
-0.00878501683473587,
0.01667439378798008,
0.043370939791202545,
-0.006380193866789341,
-0.06136675551533699,
-0.06002453342080116,
-0.020229505375027657,
0.05535770580172539,
-0.11373624950647354,
-0.04130934178829193,
0.08239977806806564,
-0.026880070567131042,
0.0007375543937087059
] |
21Quadratics
x = β (β7) Β± β ______________ (β7) 2 β 4 (3) (β1) _______________________ 2 Γ 3
x =
7 Β± β _______ 49 + 12 _______________ 6
x =
Β 7 Β± β ___ 61 ________ 6 Β
The
n x =
7 + β ___ 61 ________ 6 or x = 7 β β ___ 61 _______ 6
Or x
= 2.47 (3 s.f.) or x = β 0.135 (3 s.f.)Example 3
Solve 3x2 β 7x β1 = 0 by using the formula.
β4 Γ 3 Γ (β1) = +12a = 3, b = β7 and c = β1.
Put brackets around any negative values.Some equations cannot be easily factorised. You can also solve quadratic equations using the
quadratic formula.
β The solutions of the equation
ax2 + bx + c = 0 are given by the formula:
x = βb Β± ββ―________ b2 β 4ac _____________ 2a You n eed to rearrange the equation
into the form ax2 + bx + c = 0 before reading off
the coefficients.Watch out
1 Solve the follo
wing equations using the quadratic formula.
Give your answers exactly, leaving them in surd form where necessary.
a x2 + 3x + 1 = 0 b x2 β 3x β 2 = 0 c x2 + 6x + 6 = 0 d x2 β 5x β 2 = 0
e 3x2 + 10x β 2 = 0 f 4x2 β 4x β 1 = 0 g 4x2 β 7x = 2 h 11x2 + 2x β 7 = 0
2 Solve the follo
wing equations using the quadratic formula.
Give your answers to three significant figures.a
x2 + 4x + 2 = 0 b x2 β 8x + 1 = 0 c x2 + 11x β 9 = 0 d x2 β 7x β 17 = 0
e 5x2 + 9x β 1 = 0 f 2x2 β 3x β 18 = 0 g 3x2 + 8 = 16x h 2x2 + 11x = 5x2 β 18
3 For each of the equa
tions below, choose a suitable method and find all of the solutions.
Where necessary, give your answers to three significant figures.a
x2 + 8x + 12 = 0 b x2 + 9x β 11 = 0
c x2 β 9x β 1 = 0 d 2x2 + 5x + 2 = 0
e (2x +
8)2 = 100 f 6x2 + 6 = 12x
g 2x2 β 11 = 7x h x = Β ββ―_______ 8x β 15 You can use any method
yo
u are confident with to solve
these equations.HintExercise 2B
|
[
-0.02378137782216072,
0.04281434044241905,
0.04006641358137131,
-0.024879485368728638,
-0.017066707834601402,
0.06194112449884415,
0.0167627464979887,
0.002885577268898487,
-0.02382928505539894,
-0.04628666862845421,
0.004448902327567339,
-0.08723258227109909,
0.012003909796476364,
0.017926689237356186,
0.012088070623576641,
-0.048758577555418015,
-0.08637327700853348,
0.0774417594075203,
-0.04057249426841736,
-0.0024364928249269724,
0.04858381301164627,
-0.062327250838279724,
-0.03382384777069092,
0.0016163899563252926,
0.09519797563552856,
0.022404838353395462,
0.0032595519442111254,
0.0050154984928667545,
0.022589879110455513,
-0.006715960334986448,
-0.006672565825283527,
0.07574234902858734,
0.08675551414489746,
-0.09933582693338394,
0.0751926451921463,
-0.0043785106390714645,
-0.0063286735676229,
0.058578986674547195,
-0.04482989385724068,
-0.06679610908031464,
0.0035828494001179934,
0.023105256259441376,
0.05964094400405884,
-0.005467541050165892,
0.02351943403482437,
-0.02463480643928051,
-0.07618551701307297,
-0.04000844061374664,
0.08888646960258484,
0.03987713158130646,
0.08052776753902435,
0.10875080525875092,
-0.10194680839776993,
0.039114050567150116,
-0.05320131406188011,
-0.039946798235177994,
-0.020576078444719315,
-0.011604713276028633,
-0.036038510501384735,
-0.03264153003692627,
0.00845347810536623,
0.038817524909973145,
-0.017840657383203506,
0.04824144393205643,
0.009601771831512451,
0.04033716768026352,
0.0004099296929780394,
-0.08537033200263977,
0.08658573031425476,
0.035953011363744736,
0.0475713275372982,
0.05128553509712219,
0.004418233875185251,
-0.002007023897022009,
0.030841002240777016,
0.009054858237504959,
-0.009891833178699017,
-0.010549450293183327,
-0.014757748693227768,
-0.07564570009708405,
-0.022245202213525772,
0.045917171984910965,
0.14646261930465698,
0.04320681095123291,
-0.008869432844221592,
0.08412876725196838,
0.11804603040218353,
0.07401951402425766,
0.014953953213989735,
0.03360626846551895,
0.025235023349523544,
-0.025633152574300766,
-0.023783894255757332,
-0.03322761133313179,
-0.006804187316447496,
-0.011582471430301666,
0.06984768807888031,
-0.030858783051371574,
-0.0008345126989297569,
0.0853412076830864,
0.03560236096382141,
-0.026169471442699432,
0.05283595994114876,
0.04518568515777588,
-0.041122809052467346,
-0.025937747210264206,
-0.12355169653892517,
-0.12034445255994797,
0.0687970221042633,
-0.026683976873755455,
-0.053633809089660645,
-0.05182097479701042,
0.03267500177025795,
0.037296947091817856,
-0.03859022259712219,
-0.057313427329063416,
0.03945354372262955,
-0.017781661823391914,
0.04197290912270546,
-0.016863197088241577,
0.058360740542411804,
0.013763505965471268,
0.09540854394435883,
-0.030423862859606743,
0.045981019735336304,
0.03804238885641098,
-0.021229537203907967,
0.04681850224733353,
-0.1574447751045227,
0.008404870517551899,
0.06083034351468086,
-0.0003741250839084387,
0.009037308394908905,
-0.041500404477119446,
0.0029826636891812086,
-0.041599079966545105,
-0.019203711301088333,
0.05537011846899986,
0.01686912216246128,
-0.023050053045153618,
-0.03275939077138901,
0.10310006141662598,
0.04732499644160271,
0.02087126299738884,
0.0013788498472422361,
-0.0060113221406936646,
-0.01225474663078785,
0.04843130335211754,
-0.012852195650339127,
0.004146608989685774,
-0.011297640390694141,
-0.025155214592814445,
-0.08298369497060776,
0.04499031975865364,
-0.0541030690073967,
-0.07963334023952484,
-0.0006061536842025816,
-0.01764320768415928,
0.017143970355391502,
0.0010550572769716382,
0.011105782352387905,
-0.017355358228087425,
-0.03479301556944847,
0.039102718234062195,
0.07302885502576828,
0.034811295568943024,
-0.0778115764260292,
-0.003658436704427004,
-0.011225509457290173,
0.03832656890153885,
-0.022974055260419846,
0.06527375429868698,
0.006573478225618601,
0.04839805141091347,
0.053434696048498154,
-0.03878895565867424,
0.04995961859822273,
0.003383394330739975,
-0.032846663147211075,
0.04573822394013405,
0.06426403671503067,
-0.0198220182210207,
0.03672570362687111,
-0.022957291454076767,
-0.007064586505293846,
-0.037690166383981705,
-0.025595147162675858,
-0.03134126588702202,
0.01079430989921093,
-0.002383597195148468,
-0.013388554565608501,
-0.022975638508796692,
0.04377564415335655,
0.013037888333201408,
-0.004816052969545126,
-0.038388315588235855,
-0.08059968054294586,
-0.04859082028269768,
-0.12606483697891235,
0.13383664190769196,
0.07885372638702393,
-0.024188607931137085,
0.04023253917694092,
-0.08474893867969513,
0.028973083943128586,
0.01617639698088169,
-0.03721962496638298,
0.014966343529522419,
-0.0237729512155056,
0.0742005705833435,
-0.05837293341755867,
-0.011697105132043362,
0.001567910541780293,
-0.07542946189641953,
-0.0052209035493433475,
-0.09099934250116348,
-0.055199865251779556,
-0.03565171733498573,
-0.08248348534107208,
-0.05906306952238083,
-0.04962277412414551,
-0.013170473277568817,
0.024202369153499603,
1.994437608216365e-32,
-0.03161998093128204,
-0.1117999330163002,
-0.09556195884943008,
0.04681782424449921,
-0.00356482551433146,
-0.024862293154001236,
0.08524486422538757,
-0.07869205623865128,
0.03274986147880554,
0.032211367040872574,
0.09805294126272202,
0.07850185036659241,
-0.05549949035048485,
-0.022933680564165115,
-0.007064766716212034,
0.009002802893519402,
-0.0021394684445112944,
-0.012419428676366806,
-0.036492686718702316,
-0.0434696190059185,
-0.00042818006477318704,
0.07234618812799454,
-0.031173238530755043,
-0.005071907304227352,
-0.011115876026451588,
0.05042020604014397,
0.059356022626161575,
-0.0730491504073143,
0.01779860444366932,
0.03746051713824272,
-0.042636699974536896,
-0.025418708100914955,
0.08886086940765381,
0.08310515433549881,
-0.03230662643909454,
-0.07954392582178116,
0.04169686883687973,
0.013999090529978275,
-0.06478481739759445,
0.011275453492999077,
0.016808880493044853,
0.054091427475214005,
0.11271157115697861,
0.006594929378479719,
-0.0010913144797086716,
-0.055959612131118774,
-0.0036707851104438305,
0.041650645434856415,
-0.03315547853708267,
0.05758580565452576,
-0.04242683947086334,
0.02098425105214119,
-0.04248259961605072,
0.06784208118915558,
0.015729015693068504,
0.036682575941085815,
-0.032395459711551666,
-0.02038387581706047,
0.004642636049538851,
-0.08900447189807892,
-0.02747783623635769,
-0.07286644726991653,
0.005113641731441021,
0.017833126708865166,
0.009582005441188812,
0.11404654383659363,
0.023593001067638397,
-0.0011845758417621255,
0.04138907790184021,
0.03996347635984421,
0.02956729754805565,
0.055535510182380676,
-0.03617364540696144,
-0.03780718147754669,
-0.09515050798654556,
0.00862161722034216,
-0.04346199706196785,
0.12874172627925873,
0.007455749437212944,
-0.06950784474611282,
-0.05914447084069252,
-0.0224916972219944,
-0.04150528460741043,
0.01006027776747942,
-0.08740073442459106,
-0.0785834863781929,
0.11852600425481796,
0.04719700291752815,
0.03845856338739395,
-0.05433810129761696,
-0.017587339505553246,
0.01162196509540081,
0.008873004466295242,
0.015928329899907112,
0.17322444915771484,
4.645266266080904e-32,
0.01983383297920227,
-0.012996881268918514,
-0.0483105406165123,
-0.043523386120796204,
0.026836909353733063,
0.06005389615893364,
-0.06993666291236877,
-0.018724564462900162,
0.013157808221876621,
-0.06816194951534271,
0.08834220468997955,
0.03828843683004379,
-0.05522517114877701,
0.027691442519426346,
-0.10970625281333923,
-0.045576442033052444,
-0.05169369652867317,
0.04539991915225983,
-0.020500723272562027,
0.037395477294921875,
0.021453892812132835,
0.029383085668087006,
-0.06595349311828613,
0.04546399414539337,
0.009393448010087013,
0.04723694175481796,
-0.05046442523598671,
0.0006521821487694979,
-0.019469475373625755,
-0.07371997088193893,
0.0813797265291214,
0.0690188854932785,
-0.005751298740506172,
-0.049032095819711685,
0.002109845168888569,
-0.011352775618433952,
0.031397394835948944,
-0.05226915329694748,
-0.0022763547021895647,
-0.028270088136196136,
-0.007233243435621262,
-0.001882502343505621,
-0.07274672389030457,
-0.019678883254528046,
0.026779387146234512,
-0.05888863652944565,
-0.06420613825321198,
-0.026037275791168213,
-0.016807258129119873,
-0.0008509500185027719,
-0.052144378423690796,
0.030387138947844505,
0.0428396537899971,
0.046057529747486115,
0.024910252541303635,
-0.04615998640656471,
-0.08172410726547241,
-0.020776385441422462,
0.06200661137700081,
-0.005448815878480673,
-0.054397858679294586,
0.0968962013721466,
-0.010859795846045017,
0.07662829011678696
] |
22
Chapter 2
Given that x is positive, solve the equation
1 __ x + 1 _____ x + 2 = 28 ____ 195 Challenge Write the equation in the form
ax2 + bx + c = 0 before using the quadratic
formula or factorising.Hint
2.2 Completing the square
It is frequently useful to rewrite quadratic expressions by completing the square:
β x2 + bx = (x + b __ 2 ) 2
β ( b __ 2 ) 2
You can draw a diagram of this process when x and b
are positive:
The original rectangle has been rearranged into the
shape of a square with a smaller square missing. The two areas shaded blue are the same.b
2
x xx x =
b
2b
x2 + bx = (x + b __ 2 ) 2
β ( b __ 2 ) 2
a x2 + 8 x = ( x + 4)2 β 42
= (x + 4)2 β 16
b x2 β 3 x = (x β 3 _ 2 ) 2 β ( 3 _ 2 ) 2
= (x β 3 _ 2 ) 2 β 9 _ 4
c 2x2 β 12 x = 2( x2 β 6 x)
= 2(( x β 3)2 β 32)
= 2(( x β 3)2 β 9)
= 2(x β 3)2 β 18Example 4
Complete the square for the expressions:
a x2 + 8x b x2 β 3x c 2x2 + 12x A quadratic expression in the
form p(x + q)2 + r where p, q and r are real
constants is in completed square form.Notation
Begin by halving the coefficient of x. Using the
rule given above, b = 8 so b __ 2 = 4.
Expand the outer bracket by multiplying 2 by 9 to
get your answer in this form.4 This trapezium has an area of 50 m2.
Show that the height of the trapezium is equal to 5( ββ―__
5 β 1) m.
(x + 10) mx m
2x m
Height must be positive. You will have to discard
the negative solution of your quadratic equation.Problem-solvingP
Be careful if b __ 2 is a fraction. Here ( 3 __ 2 ) 2
= 3 2 __ 2 2 = 9 __ 4 .
Here the coefficient of x2 is 2, so take out a factor
of 2. The other factor is in the form (x2 + bx) so
you can use the rule to complete the square.
|
[
0.04041701555252075,
0.09488505125045776,
-0.007878588512539864,
-0.02522948756814003,
-0.05168547108769417,
0.01802956871688366,
0.03804917633533478,
0.046122994273900986,
-0.10012928396463394,
0.05132853984832764,
0.020967621356248856,
-0.11701325327157974,
0.02437373623251915,
-0.05348747596144676,
-0.0030166299548000097,
-0.009426314383745193,
-0.024020882323384285,
0.03548023849725723,
-0.026710493490099907,
-0.015566729940474033,
0.035227734595537186,
-0.15680594742298126,
-0.020303940400481224,
0.003854655660688877,
0.040022846311330795,
-0.0730527937412262,
-0.016939330846071243,
-0.053324222564697266,
-0.025856416672468185,
-0.0029875212348997593,
-0.12705472111701965,
0.009888572618365288,
0.04882029816508293,
-0.05417141690850258,
0.025721535086631775,
0.024250173941254616,
0.06812923401594162,
0.020456252619624138,
-0.02027886174619198,
-0.05306776985526085,
-0.02188381366431713,
0.048935744911432266,
-0.09824034571647644,
0.07124526053667068,
0.00494718411937356,
-0.12233368307352066,
-0.009717614389955997,
-0.08824783563613892,
0.04750446230173111,
-0.012372883036732674,
0.031055407598614693,
0.07973375916481018,
-0.01683802157640457,
0.029898690059781075,
-0.07777070254087448,
0.0237126424908638,
-0.07846129685640335,
0.037098273634910583,
-0.004496303852647543,
0.030771195888519287,
0.035261768847703934,
0.005336424335837364,
0.08506419509649277,
0.06159878522157669,
0.028141777962446213,
-0.002763352356851101,
-0.0628737211227417,
0.0401533804833889,
0.047698765993118286,
0.047439850866794586,
0.014728737063705921,
0.0017360615311190486,
-0.012189993634819984,
0.0029328768141567707,
0.05129053816199303,
0.005579161457717419,
-0.06797507405281067,
-0.014853608794510365,
0.04124833643436432,
-0.014269479550421238,
0.03719499334692955,
-0.012218431569635868,
0.11390717327594757,
-0.004797853529453278,
-0.037814002484083176,
-0.035642433911561966,
0.08838441967964172,
0.035896167159080505,
0.0009498331928625703,
0.007226874586194754,
0.03347005322575569,
-0.04296225309371948,
0.021903790533542633,
-0.05036807805299759,
-0.05192509666085243,
-0.09822501242160797,
0.04423815757036209,
-0.010730721056461334,
-0.018243495374917984,
0.021828416734933853,
-0.024348223581910133,
-0.01876095123589039,
0.01917875185608864,
-0.02791971154510975,
-0.04673592001199722,
-0.07463642209768295,
-0.0500870905816555,
-0.061633650213479996,
0.10353351384401321,
-0.06942539662122726,
-0.13149277865886688,
-0.0424405112862587,
-0.006051153410226107,
0.028301013633608818,
0.017547907307744026,
-0.048155199736356735,
0.07814686745405197,
-0.0659765973687172,
0.005291610956192017,
0.00714823929592967,
0.08514541387557983,
-0.014118303544819355,
0.07307501137256622,
-0.13341669738292694,
0.013178033754229546,
0.0399659126996994,
0.011742711998522282,
0.010187995620071888,
-0.011070076376199722,
-0.04899515584111214,
0.04955697059631348,
-0.013467463664710522,
-0.017812708392739296,
-0.05918343365192413,
0.05663765221834183,
-0.057043228298425674,
0.03638230636715889,
0.08016572147607803,
0.018163925036787987,
-0.06964118778705597,
0.01098490972071886,
0.07718319445848465,
0.0023923902772367,
-0.025017084553837776,
0.034462589770555496,
-0.03450284153223038,
-0.039786141365766525,
-0.0173469390720129,
-0.03977614641189575,
0.019240079447627068,
-0.03534680977463722,
-0.03170449286699295,
-0.01566382497549057,
0.03927943855524063,
0.011557570658624172,
-0.06299274414777756,
-0.03598550707101822,
-0.02507913112640381,
-0.024127300828695297,
0.004772826097905636,
-0.09217135608196259,
0.02132396772503853,
-0.04967101663351059,
-0.00918617658317089,
0.01737871579825878,
0.018725918605923653,
-0.1340605765581131,
0.02500668168067932,
-0.002746121259406209,
0.020033929497003555,
0.006017855368554592,
-0.042428746819496155,
0.01484130322933197,
0.09318374842405319,
0.02292237989604473,
-0.03366878256201744,
0.033731456845998764,
0.06293684989213943,
0.02765055000782013,
0.1050168052315712,
0.0158061683177948,
-0.010423810221254826,
-0.006111843511462212,
0.01394157949835062,
0.013063750229775906,
-0.016905445605516434,
0.05551331862807274,
-0.05184338986873627,
0.04035867005586624,
-0.010218620300292969,
-0.026009000837802887,
-0.020006339997053146,
-0.02961425483226776,
0.07452389597892761,
-0.05599922686815262,
0.02140609547495842,
-0.08838274329900742,
-0.029022080823779106,
-0.08309383690357208,
0.03537260740995407,
-0.008696584030985832,
-0.10466136038303375,
0.05687374994158745,
-0.05974643677473068,
0.03289319574832916,
-0.008052782155573368,
-0.0636848658323288,
0.002646306063979864,
-0.01751278154551983,
0.011049129068851471,
-0.0535065121948719,
0.011743416078388691,
0.005323473829776049,
-0.049574051052331924,
-0.04071572422981262,
-0.0842876136302948,
-0.008173459209501743,
0.0007610396714881063,
-0.040531910955905914,
0.049289871007204056,
-0.009467341005802155,
0.07305853813886642,
0.05064156651496887,
-4.09039120244333e-33,
-0.027145441621541977,
-0.002155940281227231,
-0.10066932439804077,
-0.017079202458262444,
-0.04237894341349602,
-0.008187452331185341,
0.0902600958943367,
-0.07725036144256592,
0.056482430547475815,
0.03598668798804283,
0.05064762011170387,
-0.006562793627381325,
-0.04375583305954933,
0.013101930730044842,
-0.03758443146944046,
-0.061659835278987885,
-0.04584049433469772,
0.02075432613492012,
-0.029594523832201958,
-0.035195428878068924,
0.0175902359187603,
0.01818682812154293,
0.07602555304765701,
-0.01808582805097103,
0.007365596480667591,
0.06801542639732361,
0.10618073493242264,
-0.0349203422665596,
0.03892679139971733,
0.09584011882543564,
-0.02226773090660572,
-0.011975282803177834,
0.09671430289745331,
0.025801226496696472,
-0.03716716170310974,
-0.08078709244728088,
0.03485585376620293,
-0.10140198469161987,
0.031821638345718384,
0.0022198616061359644,
0.050735875964164734,
0.005646937992423773,
0.12447238713502884,
0.06581788510084152,
-0.026926331222057343,
-0.019277891144156456,
0.032317984849214554,
0.019232500344514847,
0.02887766622006893,
0.05171365290880203,
-0.07491696625947952,
0.02679126150906086,
0.034266360104084015,
-0.005928629077970982,
0.021062785759568214,
0.03551117330789566,
-0.055909886956214905,
-0.042041465640068054,
0.03903613239526749,
-0.06335591524839401,
0.024335244670510292,
-0.01844603940844536,
0.05099202319979668,
0.06850302964448929,
0.005790320690721273,
0.04220245033502579,
0.018361693248152733,
-0.029003845527768135,
0.008230788633227348,
0.07195726037025452,
-0.0357656367123127,
0.0477009117603302,
-0.11138997972011566,
-0.023017607629299164,
-0.02210834063589573,
0.05309789255261421,
-0.06852050125598907,
0.07736135274171829,
0.016173910349607468,
-0.002166004152968526,
-0.08383328467607498,
0.12445586174726486,
-0.03468233346939087,
-0.04242297634482384,
-0.030276501551270485,
-0.013889291323721409,
0.10259737819433212,
0.02671636827290058,
0.07934418320655823,
-0.03370344638824463,
-0.030739150941371918,
0.02113105170428753,
-0.023214690387248993,
-0.04277990758419037,
0.08665594458580017,
8.080872438032871e-32,
-0.02242773026227951,
-0.039581749588251114,
-0.008581498637795448,
-0.015504593029618263,
-0.0010125659173354506,
0.06695046275854111,
-0.026350127533078194,
-0.010321107693016529,
-0.01430612150579691,
-0.077647864818573,
-0.01640029065310955,
0.06200674921274185,
-0.03223317861557007,
0.08171428740024567,
-0.10687325894832611,
-0.02679675817489624,
-0.04595496878027916,
0.06951982527971268,
-0.009111979976296425,
-0.03046300821006298,
0.025861958041787148,
0.027748405933380127,
-0.04285769164562225,
0.09058581292629242,
-0.02084188163280487,
0.05212625488638878,
-0.024651331827044487,
0.01567714475095272,
0.011494284495711327,
-0.07050123810768127,
0.1176571324467659,
0.05675498768687248,
-0.020929815247654915,
0.05086204782128334,
0.03146100416779518,
-0.049740638583898544,
0.018766148015856743,
-0.02134949341416359,
0.010243088938295841,
-0.02291036956012249,
-0.030863115563988686,
-0.0063031320460140705,
-0.07460449635982513,
0.0022118939086794853,
-0.023431744426488876,
-0.0016832291148602962,
-0.03509008139371872,
-0.049563899636268616,
0.006531321443617344,
0.016520865261554718,
-0.022937241941690445,
0.018040252849459648,
0.1325162947177887,
0.03406514227390289,
0.024403085932135582,
-0.054889120161533356,
-0.140156090259552,
-0.013686250895261765,
0.0377788245677948,
-0.01308264397084713,
-0.1072104275226593,
0.10502892732620239,
-0.056975264102220535,
-0.031087983399629593
] |
23Quadratics
3x2 + 6 x + 1
= 3(x2 + 2x) + 1
= 3(( x + 1)2 β 12) + 1
= 3(x + 1)2 β 3 + 1
= 3(x + 1)2 β 2
So p = 3, q = 1 and r = β 2.Example 5
Write 3x2 + 6x + 1 in the form p(x + q)2 + r, where p, q and r are integers to be found.
1 Complete the square for the e
xpressions:
a x2 + 4x b x2 β 6x c x2 β 16x d x2 + x e x2 β 14
2 Complete the square for the e
xpressions:
a 2x2 + 16x b 3x2 β 24x c 5x2 + 20x d 2x2 β 5x e 8x β 2x2
3 Write each of these expressions in the form p(x + q)2 + r, where p, q and r are constants
to be found:
a 2x2 + 8x + 1 b 5x2 β 15x + 3 c 3x2 + 2x β 1 d 10 β 16 x β 4x2 e 2x β 8x2 + 10
4 Given tha
t x2 + 3x + 6 = (x + a)2 + b, find the values of the constants a and b. (2 marks)
5 Write 2
+ 0.8x β 0.04x2 in the form A β B(x + C)2, where A, B and C are constants to
beΒ determined. (3 marks)E
EExercise 2C
Solve the equation x2 + 8x + 10 = 0 by completing the square.
Give your answers in surd form.Example 6
x2 + 8 x + 10 = 0
x2 + 8 x = β10
(x + 4)2 β 42 = β10
(x + 4)2 = β10 + 16
(x + 4)2 = 6
(x + 4) = Β± ββ―__
6
x =
β4 Β± ββ―__
6
So th
e solutions are
x = β4 + ββ―__
6 and x = β 4 β ββ―__
6 .Check coefficient of x2 = 1.
Subtract 10 to get the LHS in the form x2 + bx.
Complete the square for x2 + 8x.
Add 42 to both sides.
Take square roots of both sides.
Subtract 4 from both sides.
Leave your answer in surd form.β a x 2 + bx + c = a (x + b ___ 2a ) 2
+ (c β b 2 ____ 4 a 2 )
You could also use the rule given above to
complete the square for this expression, but it is safer to learn the method shown here. This is an expression , so you canβt
divide every term by 3 without changing its value. Instead, you need to take a factor of 3 out of 3x
2 + 6x .Watch out
In question 3d ,
wr
ite the expression as
β4x2 β 16 x + 10 then
take a factor of β 4 out
of the first two terms to get β 4(x
2 + 4x) + 10.Hint
|
[
-0.01306331530213356,
0.07446865737438202,
-0.008622128516435623,
-0.047031763941049576,
0.03577741980552673,
0.07726112008094788,
0.008979957550764084,
-0.036023955792188644,
-0.05742703005671501,
0.10164717584848404,
-0.05958917737007141,
0.0121842036023736,
0.0020413044840097427,
-0.0058693778701126575,
0.08339106291532516,
-0.026423990726470947,
-0.07100033760070801,
0.02865041233599186,
-0.06570331007242203,
-0.06150703504681587,
0.07634977996349335,
-0.08819306641817093,
-0.05711128190159798,
-0.021331701427698135,
0.09596212208271027,
-0.025834089145064354,
0.027980608865618706,
-0.05427539721131325,
-0.01899252086877823,
-0.02088155411183834,
0.005948690231889486,
0.03970034793019295,
0.1097770482301712,
-0.06828714907169342,
0.0631813257932663,
0.02928190305829048,
0.022966409102082253,
0.06427936255931854,
-0.03426570072770119,
-0.017794454470276833,
-0.03263962268829346,
0.02323986403644085,
-0.08876325935125351,
-0.03432333841919899,
0.09225886315107346,
-0.07526372373104095,
0.010032230988144875,
-0.011844542808830738,
0.04910803586244583,
-0.020861295983195305,
0.07413864135742188,
0.028803449124097824,
-0.03333813324570656,
0.07195397466421127,
-0.009564109146595001,
-0.0052544423379004,
-0.05993634834885597,
0.07168242335319519,
-0.06991151720285416,
0.01863875985145569,
-0.02819293923676014,
0.03369217738509178,
-0.011847774498164654,
0.0774465873837471,
-0.028912454843521118,
0.03384624794125557,
-0.05798428878188133,
-0.02296566218137741,
0.04426538199186325,
0.0351078137755394,
0.04068060964345932,
0.07417068630456924,
-0.050102975219488144,
-0.016135113313794136,
0.0442952960729599,
0.05549563840031624,
-0.029797274619340897,
-0.07088145613670349,
-0.05481354519724846,
0.012028345838189125,
-0.08098729699850082,
0.03188946470618248,
0.005004630424082279,
0.06989585608243942,
-0.021423624828457832,
0.12959912419319153,
0.06871002167463303,
0.03415672853589058,
0.021885208785533905,
0.011615391820669174,
-0.0022806604392826557,
-0.0005227802321314812,
-0.022705361247062683,
-0.09526390582323074,
-0.0015980523312464356,
-0.03874476999044418,
0.055190738290548325,
-0.09905924648046494,
-0.0501173660159111,
0.05991910398006439,
0.08722212165594101,
0.03009197674691677,
-0.047869086265563965,
0.025483854115009308,
-0.06740536540746689,
-0.014560543932020664,
0.01758350059390068,
-0.03893982991576195,
0.06578635424375534,
-0.07811400294303894,
-0.1042637825012207,
-0.07345771789550781,
0.03900991380214691,
0.04161612316966057,
0.018865881487727165,
-0.04853856563568115,
0.002354560187086463,
-0.01997203752398491,
0.018943198025226593,
-0.06702244281768799,
0.06495581567287445,
-0.039237845689058304,
0.03336594998836517,
-0.006459445226937532,
0.01577191986143589,
-0.005448475480079651,
-0.013944143429398537,
-0.005587069317698479,
-0.047305379062891006,
-0.0025142929516732693,
0.0004466598038561642,
-0.0007264427840709686,
-0.0017058160156011581,
-0.037212058901786804,
0.04164395481348038,
0.0243754293769598,
0.015975093469023705,
0.017917361110448837,
-0.07513222843408585,
-0.04058380052447319,
0.06347247958183289,
0.10238227248191833,
-0.015793602913618088,
-0.028043998405337334,
0.03882380947470665,
-0.020289309322834015,
0.012157632037997246,
0.017168322578072548,
-0.0855100005865097,
0.06964728981256485,
0.03552419692277908,
-0.04915099963545799,
-0.01860574446618557,
0.08254632353782654,
0.03258559852838516,
-0.03846624493598938,
-0.02252092957496643,
-0.04329252243041992,
-0.00942728016525507,
-0.018367990851402283,
-0.13035078346729279,
-0.027700120583176613,
-0.08071114122867584,
0.08294083178043365,
0.0425872802734375,
0.037221282720565796,
-0.03986799344420433,
0.031183885410428047,
0.021950600668787956,
-0.002188708633184433,
-0.035546351224184036,
0.052243005484342575,
-0.0063347965478897095,
0.010471079498529434,
-0.005786361638456583,
-0.0596720390021801,
0.09611000865697861,
-0.018906816840171814,
0.009054108522832394,
0.13768580555915833,
0.06838382035493851,
0.022195765748620033,
-0.005149372853338718,
-0.028657516464591026,
0.01283933874219656,
0.07036841660737991,
0.01765136420726776,
-0.007352069020271301,
-0.015151066705584526,
0.006643138360232115,
0.046536222100257874,
-0.10215692967176437,
0.032457660883665085,
0.02978210151195526,
-0.05231979116797447,
-0.014164036139845848,
-0.019782627001404762,
0.025463048368692398,
-0.0786454901099205,
-0.020982148125767708,
0.04996363818645477,
0.008014521561563015,
0.08325330913066864,
-0.07927465438842773,
-0.01689421385526657,
-0.003823251463472843,
-0.062080882489681244,
0.012011546641588211,
0.021015634760260582,
0.06137191876769066,
-0.036454781889915466,
0.030463766306638718,
0.056925948709249496,
-0.10334885120391846,
-0.06746993213891983,
-0.04054517298936844,
0.04065193235874176,
-0.07646400481462479,
-0.05776436999440193,
0.030676426365971565,
-0.006534946616739035,
-0.021049782633781433,
0.024285167455673218,
-1.0797121323814105e-32,
-0.053499508649110794,
-0.028041522949934006,
-0.08376973867416382,
-0.07937971502542496,
-0.023712845519185066,
-0.025061296299099922,
-0.02109374850988388,
-0.11313527077436447,
0.10869971662759781,
0.018136009573936462,
0.04307224974036217,
-0.021633004769682884,
0.016455966979265213,
0.023688873276114464,
-0.04969646409153938,
-0.0069916886277496815,
0.008258311077952385,
0.04616459831595421,
-0.01869288645684719,
-0.0862007588148117,
0.045924972742795944,
0.024555524811148643,
-0.03236589580774307,
-0.012139721773564816,
0.033840399235486984,
0.1062992513179779,
-0.01707318425178528,
-0.11590728163719177,
0.03173452615737915,
0.02118624933063984,
-0.03896276652812958,
-0.030048465356230736,
0.026364197954535484,
0.02604546770453453,
-0.030245942994952202,
-0.08196042478084564,
0.08082358539104462,
0.003454421181231737,
0.013181690126657486,
0.008829881437122822,
0.06160704046487808,
-0.08466995507478714,
0.00496568763628602,
0.08596986532211304,
0.012687554582953453,
-0.02956209145486355,
0.020818037912249565,
0.03605632483959198,
-0.01175547856837511,
-0.008724065497517586,
-0.08789769560098648,
0.0013664447469636798,
-0.0771118775010109,
-0.010675091296434402,
0.01816759631037712,
0.006408455315977335,
-0.001488559995777905,
-0.010462544858455658,
0.03894289955496788,
0.02127055637538433,
-0.055239856243133545,
0.025471044704318047,
-0.007308585569262505,
0.0200158953666687,
-0.0021421259734779596,
0.02463151142001152,
-0.0009927648352459073,
-0.007253666874021292,
-0.011119821108877659,
0.07124841213226318,
-0.01672211103141308,
0.004641491919755936,
-0.04544344171881676,
-0.11203368008136749,
-0.02688831090927124,
0.048858851194381714,
-0.07394713163375854,
0.14667755365371704,
-0.0296147633343935,
-0.021371861919760704,
-0.07524668425321579,
-0.00015761354006826878,
0.02612259052693844,
-0.011444106698036194,
-0.09659101814031601,
0.015112055465579033,
0.1005038246512413,
0.08732079714536667,
-0.00496472604572773,
0.006034768186509609,
-0.09123006463050842,
-0.0036753362510353327,
0.016985982656478882,
0.05865481123328209,
0.09573464840650558,
8.915166035343955e-32,
0.010435404255986214,
-0.04008499160408974,
-0.09216891974210739,
-0.057214315980672836,
-0.019288314506411552,
-0.008920987136662006,
0.003557814285159111,
0.035217516124248505,
-0.03843192383646965,
-0.09618530422449112,
0.0015115565620362759,
-0.04434444010257721,
-0.021375106647610664,
-0.0055406722240149975,
-0.1187179759144783,
-0.04668695107102394,
-0.01162967924028635,
0.031193243339657784,
-0.037657905369997025,
-0.004443871323019266,
0.0072599793784320354,
-0.03623674064874649,
-0.04644004628062248,
0.029543347656726837,
0.017195967957377434,
0.07695747911930084,
0.0034644347615540028,
-0.07326814532279968,
0.02189468778669834,
0.005187151487916708,
0.057620350271463394,
0.0632769837975502,
-0.03708403930068016,
-0.01786363683640957,
0.04325209558010101,
-0.041024766862392426,
0.111813023686409,
-0.07668143510818481,
-0.06389899551868439,
-0.04208047315478325,
0.028731778264045715,
-0.0984559953212738,
-0.08541138470172882,
0.044769056141376495,
0.017212143167853355,
0.026696069166064262,
0.01994369737803936,
-0.07507597655057907,
0.02878292091190815,
-0.04356461018323898,
-0.07344082742929459,
0.04334196075797081,
0.06649413704872131,
-0.035794928669929504,
-0.038787588477134705,
-0.05311178043484688,
-0.07900669425725937,
0.035844676196575165,
0.024630846455693245,
-0.045692767947912216,
-0.04764474928379059,
0.09080272167921066,
-0.03625921905040741,
-0.01324811577796936
] |
24
Chapter 2
Solve the equation 2x2 β 8x + 7 = 0. Give your answers in surd form.Example 7
2x2 β 8 x + 7 = 0
x2 β 4 x + 7 __ 2 = 0
x2 β 4 x = β 7 __ 2
(x β
2)2 β 22 = β 7 __ 2
(x β
2)2 = β 7 __ 2 + 4
(x
β 2)2 = 1 __ 2
x β
2 = Β± ββ―__
1 __ 2
x =
2 Β± 1 ___ ββ―__
2
So th
e roots are
x = 2 + 1 ___ ββ―__
2 and x = 2 β 1 ___ ββ―__
2 Complete the square for x2 β 4x.
Add 22 to both sides.
Take square roots of both sides.
Add 2 to both sides.
1 Solve these quadratic equations by completing the square. Leave your answers in surd form.
a x2 + 6x + 1 = 0 b x2 + 12x + 3 = 0 c x2 + 4x β 2 = 0 d x2 β 10x = 5
2 Solve these quadra
tic equations by completing the square. Leave your answers in surd form.
a 2x2 + 6x β 3 = 0 b 5x2 + 8x β 2 = 0 c 4x2 β x β 8 = 0 d 15 β 6 x β 2x2 = 0
3 x2 β 14x + 1 = (x + p)2 + q, where p and q are constants.
a Find the values of
p and q. (2 marks)
b Using your answ
er to part a, or otherwise, show that the solutions to the equation
x2 β 14x + 1 = 0 can be written in the form r Β± s ββ―__
3 , where r and s are constants
to be found. (2 marks)
4 By completing the square, sho
w that the solutions to
the equation x2 + 2bx + c = 0 are given by the formula
x = β b Β± ββ―______ b 2 β c . (4 marks)E
E/P
Follow the same steps as you would
if the coefficients were numbers.Problem-solvingExercise 2D
Start by dividing the whole
equation by a .Hint
You can use this
methodΒ to prove the quadratic
formula. β Section 7.4Linksa Show that the solutions to the equation
ax2 + 2bx + c = 0 are given by x = β b __ a Β± ββ―______ b 2 β ac ______ a 2 .
b Hence, or otherwise, sho
w that the solutions to the
equation ax2 + bx + c = 0 can be written as
x = βb Β± ββ―_______ b 2 β 4ac ____________ 2a .Challenge Use your calculator to check
so
lutions to quadratic equations quickly.OnlineThis is an equation so you can divide every term
by the same constant. Divide by 2 to get x2 on its
own. The right-hand side is 0 so it is unchanged.Problem-solving
|
[
0.013856290839612484,
0.0842919647693634,
0.0033482795115560293,
0.03280477598309517,
0.013162902556359768,
0.05939527601003647,
-0.0175707396119833,
0.11280366778373718,
-0.04327024146914482,
0.02562318556010723,
0.060167163610458374,
-0.07456247508525848,
-0.017832111567258835,
-0.07127248495817184,
-0.0026145954616367817,
-0.08568552881479263,
-0.04192982614040375,
0.026245519518852234,
-0.019915705546736717,
-0.013538878411054611,
0.07562603056430817,
-0.0443815179169178,
-0.01412374246865511,
-0.004404843784868717,
0.07991339266300201,
-0.05551699921488762,
-0.026060475036501884,
-0.059739310294389725,
0.024286258965730667,
-0.03461470082402229,
0.023910244926810265,
0.03370067477226257,
0.0606442354619503,
-0.0009170433040708303,
0.08060666173696518,
0.03856939077377319,
0.05868428573012352,
-0.012628581374883652,
-0.06163390353322029,
-0.10228496044874191,
-0.058986272662878036,
-0.015031016431748867,
-0.0022662717383354902,
-0.04971710219979286,
0.07441779226064682,
-0.0662311241030693,
-0.09185580909252167,
0.00035436401958577335,
0.06672146916389465,
-0.04927738383412361,
0.035204119980335236,
0.042216844856739044,
0.0076749976724386215,
0.02499682828783989,
0.01644725725054741,
-0.009115130640566349,
-0.064357228577137,
0.0026633592788130045,
-0.021860791370272636,
0.009156832471489906,
-0.012312768027186394,
-0.07208480685949326,
-0.004992786329239607,
0.047547854483127594,
-0.018348336219787598,
0.0518127977848053,
-0.02478197030723095,
-0.05473433434963226,
0.03876815736293793,
0.0077250306494534016,
-0.020133113488554955,
0.04804108664393425,
-0.050348080694675446,
-0.03507103770971298,
0.05509715899825096,
-0.048006195574998856,
-0.04768171161413193,
-0.04855343699455261,
0.055718377232551575,
-0.011693672277033329,
0.03498039022088051,
-0.04702194035053253,
0.0871993899345398,
-0.030497567728161812,
0.019984859973192215,
-0.07949892431497574,
0.03627161681652069,
-0.034223880618810654,
0.052087631076574326,
0.017224131152033806,
0.10746700316667557,
-0.004580747336149216,
0.0013586576096713543,
-0.04112763702869415,
0.014267859049141407,
-0.07723904401063919,
0.0417327955365181,
-0.03719297796487808,
0.03419267013669014,
0.09393253177404404,
0.057810038328170776,
-0.08989057689905167,
-0.03088526427745819,
0.04446650668978691,
-0.03478265553712845,
-0.09357313066720963,
-0.042948197573423386,
-0.04906278848648071,
0.09753719717264175,
-0.11823976039886475,
-0.09541834890842438,
-0.057047951966524124,
-0.027110183611512184,
-0.03633323684334755,
0.025452714413404465,
-0.041114650666713715,
0.11075125634670258,
-0.02974766492843628,
-0.07025765627622604,
-0.06954250484704971,
0.02205638214945793,
-0.005614385474473238,
0.1076149120926857,
-0.05126793310046196,
-0.0019355334807187319,
-0.015597830526530743,
0.05538295581936836,
-0.008635505102574825,
-0.07313027232885361,
-0.09371698647737503,
-0.008611028082668781,
-0.05005380138754845,
-0.08259439468383789,
0.0006331573240458965,
0.003663737792521715,
-0.0012875794200226665,
-0.033511314541101456,
0.1003296971321106,
0.0023040277883410454,
-0.063785120844841,
0.010816617868840694,
0.04572944715619087,
0.028402220457792282,
0.027244767174124718,
0.03746296465396881,
0.006191635970026255,
-0.043872661888599396,
0.03588500991463661,
0.03788438066840172,
0.04856051877140999,
-0.061570774763822556,
-0.037816718220710754,
-0.04307584837079048,
0.01976826973259449,
-0.01760963164269924,
-0.05710715427994728,
-0.04065122455358505,
-0.09205695241689682,
-0.0015137698501348495,
-0.052526041865348816,
-0.08528895676136017,
0.04219955578446388,
-0.025699060410261154,
0.07889733463525772,
0.07277151197195053,
0.04144468531012535,
-0.0871797576546669,
0.04249408468604088,
-0.01959129609167576,
-0.05450783669948578,
0.033690523356199265,
0.04691562056541443,
0.04562748968601227,
0.05436078831553459,
0.06131155043840408,
-0.06899310648441315,
0.11461915820837021,
-0.01944645307958126,
0.009938712231814861,
0.07970456033945084,
0.005708109587430954,
0.03625224530696869,
0.011043674312531948,
-0.026771221309900284,
0.04062642902135849,
-0.061838310211896896,
-0.00575590506196022,
0.04970255494117737,
0.020446283742785454,
0.014156853780150414,
-0.047625668346881866,
-0.01670818030834198,
-0.01559675857424736,
0.05560470372438431,
-0.09243116527795792,
-0.03853324428200722,
0.05225929617881775,
-0.043513402342796326,
-0.1449379026889801,
0.09706253558397293,
0.04667843133211136,
-0.01721479371190071,
0.04861126095056534,
-0.06244334951043129,
0.04208045452833176,
0.05278753489255905,
-0.04779873415827751,
0.01838296838104725,
0.010512755252420902,
0.07263839989900589,
-0.024861332029104233,
0.020554296672344208,
-0.03367958217859268,
-0.052438609302043915,
-0.03143769130110741,
-0.0914146825671196,
0.013937108218669891,
-0.0786006897687912,
-0.033623579889535904,
-0.031579114496707916,
-0.08400172740221024,
-0.09520289301872253,
0.08071447163820267,
-2.9025152226870378e-33,
-0.030630027875304222,
0.000723019358702004,
-0.13082700967788696,
-0.0222557932138443,
-0.07064206898212433,
-0.003852914785966277,
0.07162008434534073,
-0.02850349247455597,
0.06118253991007805,
0.033624086529016495,
0.030085185542702675,
0.05358497053384781,
-0.015535727143287659,
0.0342051237821579,
-0.049156371504068375,
-0.011623857542872429,
-0.04763619229197502,
-0.029999492689967155,
0.04321658983826637,
0.03564636781811714,
-0.01168028637766838,
0.06932375580072403,
0.03923272341489792,
-0.037968773394823074,
0.035295091569423676,
0.04498608782887459,
0.07554447650909424,
-0.03995182365179062,
0.026151517406105995,
0.07222757488489151,
-0.022064007818698883,
-0.07509538531303406,
0.0657639354467392,
0.012735869735479355,
-0.058937206864356995,
0.004727000370621681,
-0.021522993221879005,
0.000836359104141593,
0.001769278896972537,
0.006531986407935619,
0.06412534415721893,
0.011980363167822361,
-0.006130211520940065,
-0.013916737399995327,
0.0175770353525877,
0.10004150867462158,
0.049749650061130524,
0.09515973180532455,
-0.01664825715124607,
-0.02298170141875744,
-0.018174275755882263,
-0.020139846950769424,
0.04000147059559822,
0.006928328890353441,
0.06574150174856186,
0.008514303714036942,
0.018739912658929825,
-0.07956217974424362,
0.05273677036166191,
0.004681436810642481,
-0.02285297028720379,
-0.0052055600099265575,
0.04431821405887604,
0.043736349791288376,
-0.010998111218214035,
-0.003987899515777826,
-0.05144169181585312,
0.01290341280400753,
0.00007037919567665085,
0.016425106674432755,
-0.042121268808841705,
0.019166233018040657,
-0.07010328769683838,
-0.057352546602487564,
-0.021360624581575394,
0.03645457327365875,
-0.04301493987441063,
0.06326465308666229,
-0.02072448842227459,
-0.13151687383651733,
-0.12605583667755127,
-0.07041509449481964,
-0.02722637914121151,
-0.05482713505625725,
-0.0808548703789711,
-0.029226554557681084,
0.10919325798749924,
0.02556312270462513,
0.038477350026369095,
-0.038776934146881104,
0.026563480496406555,
0.031170666217803955,
0.016131317242980003,
-0.0032290148083120584,
0.09234629571437836,
9.094410115934741e-32,
-0.015117178671061993,
0.011422949843108654,
0.034076888114213943,
-0.021036265417933464,
0.056989189237356186,
0.07185154408216476,
0.0004755224799737334,
0.0029015971813350916,
0.009531001560389996,
-0.044730283319950104,
0.008366508409380913,
0.015945369377732277,
-0.05620722100138664,
0.0644788146018982,
0.0001901404611999169,
-0.0012804720317944884,
-0.09175882488489151,
0.03629952669143677,
0.001254928414709866,
-0.0047879875637590885,
-0.007271636743098497,
-0.00941763911396265,
-0.08714170753955841,
0.05191091075539589,
0.0076904878951609135,
0.08347585797309875,
-0.07597760856151581,
-0.041225139051675797,
-0.023290690034627914,
0.010585757903754711,
0.07407207787036896,
-0.02242426574230194,
-0.05224563553929329,
-0.04014478623867035,
0.02833976410329342,
-0.01695319265127182,
0.06502744555473328,
0.011213625781238079,
-0.011792244389653206,
0.057186782360076904,
-0.01202533021569252,
0.015157723799347878,
-0.03131554275751114,
0.0730736181139946,
-0.03778310865163803,
-0.053030405193567276,
-0.008550223894417286,
-0.008784736506640911,
0.037109456956386566,
-0.015219314023852348,
-0.06008302420377731,
0.019612034782767296,
0.05267752334475517,
0.05318663641810417,
0.032592158764600754,
-0.11173870414495468,
-0.08933089673519135,
0.053916141390800476,
0.04421485215425491,
-0.07695169746875763,
-0.0703887790441513,
0.03970501571893692,
-0.04392121732234955,
0.06746881455183029
] |
25Quadratics
2.3 Functions
A function is a mathematical relationship that maps each value of a set of inputs to a single output.
The notation f(x) is used to represent a function of x.
β The set of possible inputs for a function is called the domain.
3DomainR ange
7
β7
2f(β7) = 49f(7) = 49f(3) = 9
f( 2) = 29
49
49
21
41
16 f( ) = 14 1
16β The set of possible outputs of a function is called the range.
This diagram sho
ws how the function f(x) = x2 maps five
values in its domain to values in its range.
β The roots of a function are the values of x for which f( x) = 0.
The functions f and g are given by f(x) = 2x β 10 If the i nput of a function,
x, can be any real number the
domain can be written as x β
β .
The s
ymbol β mean
s βis a member
ofβ and the symbol β rep
resents the
real numbers.Notation
and g(x) = x2 β 9, xΒ β β .
a Find the values of
f(5) and g(10).
b Find the value of
x for which f(x) = g(x).Example 8
a f(5) = 2(5) β 10 = 10 β 10 = 0
g
(10) = (10)2 β 9 = 100 β 9 = 91
b f(x) =
g(x)
2x β 10 = x2 β 9
x2 β 2 x + 1 = 0
(x β 1)2 = 0
x = 1To find f(5), substitute x = 5 into the function f(x).
Set f(x) equal to g(x) and solve for x.
The function f is defined as f(x) = x2 + 6x β 5, Β x β β .
a Write f(x
) in the form (x + p)2 + q.
b Hence, or otherwise, find the r
oots of f(x), leaving your answers in surd form.
c Write down the minim
um value of f(x), and state the value of x for which it occurs.Example 9
a f(x) = x2 + 6 x β 5
= (x + 3)2 β 9 β 5
= (x + 3)2 β 14
b f(x) =
0
(x + 3)2 β 14 = 0
(x + 3)2 = 14
x + 3 = Β± ββ―____ 14
x = β3 Β± ββ―____ 14
f(x) has two roots:
β3 + ββ―____ 14 and β 3 β ββ―____ 14 .Complete the square for x2 + 6x and then
simplify the expression.
You can solve this equation directly. Remember to
write Β± when you take square roots of both sides.To find the root(s) of a function, set it equal to zero.
|
[
0.047365691512823105,
0.0573720708489418,
0.005755032878369093,
-0.06053665652871132,
-0.10266956686973572,
0.06456391513347626,
-0.010251613333821297,
0.01822834275662899,
0.07105199247598648,
-0.04918083921074867,
0.01122002862393856,
-0.006405067630112171,
0.014039368368685246,
0.10179648548364639,
0.013017594814300537,
-0.013301519677042961,
0.02622189372777939,
0.047937966883182526,
-0.039125096052885056,
-0.095669686794281,
0.07618044316768646,
0.054133396595716476,
-0.13757820427417755,
-0.04437040537595749,
-0.01150733046233654,
-0.03032810054719448,
-0.03700344264507294,
-0.003193049458786845,
-0.04857226088643074,
-0.046104684472084045,
-0.07071144133806229,
0.06855112314224243,
0.009701743721961975,
-0.03686673939228058,
-0.020847104489803314,
0.012644021771848202,
-0.01902620680630207,
8.607512995695288e-7,
0.013444267213344574,
-0.021859275177121162,
0.02563447318971157,
0.024840759113430977,
0.007585132494568825,
-0.017414597794413567,
-0.017464986070990562,
0.01485507097095251,
0.05141512304544449,
-0.02983294613659382,
0.03852158039808273,
0.016105040907859802,
-0.02026280015707016,
0.12681759893894196,
-0.16436630487442017,
0.08102858811616898,
0.05139411613345146,
0.004181805066764355,
0.038486141711473465,
-0.11066220700740814,
-0.06276047229766846,
0.033241186290979385,
-0.05346842110157013,
0.04524088650941849,
0.01562909595668316,
0.01857614517211914,
0.05287153273820877,
0.06446517258882523,
-0.004196339286863804,
-0.059773270040750504,
-0.01211139839142561,
0.053073108196258545,
-0.10148820281028748,
-0.017448317259550095,
-0.008673873730003834,
0.01886359415948391,
0.07333435863256454,
0.038382191210985184,
-0.03300065919756889,
-0.015844611451029778,
0.008209056220948696,
-0.03726814687252045,
0.007934979163110256,
0.009686875157058239,
0.012043701484799385,
0.08179301023483276,
0.02860109880566597,
0.04779980704188347,
0.010313940234482288,
0.02818593755364418,
-0.006014091894030571,
0.03930428996682167,
-0.014907141216099262,
0.044264428317546844,
-0.034317661076784134,
-0.022903472185134888,
0.0023655968252569437,
-0.0427866205573082,
0.083876833319664,
-0.11145688593387604,
0.02133963443338871,
0.04343399778008461,
-0.050012923777103424,
0.03881412744522095,
0.04110635817050934,
-0.05305726081132889,
-0.06539316475391388,
-0.027036089450120926,
-0.024370769038796425,
-0.08405718207359314,
-0.03869396448135376,
-0.08421167731285095,
0.021999554708600044,
-0.10614190250635147,
-0.03907616063952446,
0.0048134117387235165,
0.02354089543223381,
-0.0846618041396141,
-0.029515134170651436,
-0.033210389316082,
0.0645669549703598,
-0.02702638879418373,
0.03399581089615822,
-0.03779807314276695,
-0.045480046421289444,
0.03098602592945099,
-0.04047093167901039,
0.00123339903075248,
0.03297240659594536,
0.02021694742143154,
-0.0311830285936594,
-0.01654639281332493,
0.05123239383101463,
-0.010141807608306408,
-0.03132276609539986,
-0.008819390088319778,
0.008005628362298012,
-0.013875487260520458,
-0.07132314145565033,
0.13618861138820648,
0.018930992111563683,
0.07764116674661636,
0.09149857610464096,
0.03148889169096947,
0.07622496783733368,
0.02847382053732872,
0.0988936498761177,
0.00666803726926446,
0.03976350650191307,
-0.044285647571086884,
0.04218960925936699,
0.005865375977009535,
0.017495106905698776,
0.030699020251631737,
0.027519017457962036,
0.10130581259727478,
-0.026434728875756264,
0.06018060818314552,
-0.03221019357442856,
-0.041304949671030045,
-0.012089155614376068,
-0.008918347768485546,
0.06217435002326965,
-0.04136253148317337,
0.02369164302945137,
-0.005797009915113449,
0.032585788518190384,
0.05069849640130997,
0.01865246146917343,
-0.007182655390352011,
-0.11293087154626846,
-0.04485929384827614,
-0.03065667301416397,
-0.0183780025690794,
0.022124525159597397,
0.016862571239471436,
-0.06572272628545761,
0.06842152774333954,
0.026338528841733932,
-0.010448187589645386,
-0.03328879177570343,
-0.012391227297484875,
0.059842549264431,
-0.06764142215251923,
0.070070281624794,
-0.019572827965021133,
-0.02624662034213543,
-0.04322746768593788,
-0.04303206875920296,
-0.019591595977544785,
-0.02047964185476303,
-0.0357670858502388,
-0.06459228694438934,
-0.032688431441783905,
-0.02429637871682644,
-0.004903712309896946,
0.00492318207398057,
-0.026281658560037613,
0.013459809124469757,
0.1030661091208458,
0.09639713168144226,
0.10669773817062378,
0.0031336965039372444,
-0.013106027618050575,
0.03332802653312683,
0.017633993178606033,
-0.06685832142829895,
0.01567990519106388,
-0.07658547908067703,
-0.044414978474378586,
-0.0007842897903174162,
0.07889304310083389,
-0.09204216301441193,
0.007281436584889889,
-0.03787679597735405,
-0.05557142570614815,
-0.04588155820965767,
-0.08936375379562378,
0.010106880217790604,
-0.017630189657211304,
-0.12588319182395935,
0.006239703856408596,
0.002452898072078824,
-0.0035609037149697542,
0.022165777161717415,
1.2633895899163224e-32,
-0.006679626181721687,
0.059772443026304245,
-0.08341804146766663,
0.017872460186481476,
0.05629129707813263,
-0.05460674688220024,
0.08760877698659897,
-0.005449151620268822,
0.1031411662697792,
0.07011615484952927,
-0.012144426815211773,
0.07374413311481476,
-0.019957760348916054,
-0.07898473739624023,
-0.084442138671875,
-0.02708260715007782,
-0.03601613640785217,
-0.10953884571790695,
-0.016420932486653328,
-0.004369511269032955,
0.07109881192445755,
0.0006118757301010191,
-0.05908465012907982,
0.019554195925593376,
-0.04485699534416199,
0.032036688178777695,
0.038657717406749725,
0.005906912498176098,
0.013369621708989143,
0.07053744792938232,
-0.07404059916734695,
0.021128449589014053,
0.11331170797348022,
0.08005213737487793,
-0.014506055042147636,
-0.026593783870339394,
0.08934549242258072,
-0.07344356179237366,
-0.052294351160526276,
0.018534265458583832,
0.07924666255712509,
-0.022656170651316643,
0.09034551680088043,
0.028965944424271584,
0.03922714293003082,
0.045719560235738754,
0.03774132952094078,
0.016077091917395592,
0.004434530157595873,
0.050955068320035934,
-0.14357705414295197,
-0.100044384598732,
-0.002646882552653551,
0.016888687387108803,
0.06665108352899551,
0.0064374227076768875,
-0.021455800160765648,
-0.03421841189265251,
0.0027843923307955265,
0.027946926653385162,
0.007137373089790344,
-0.0560801662504673,
0.02197265811264515,
0.05364140123128891,
-0.049335792660713196,
0.05883794277906418,
-0.04508443549275398,
0.036824747920036316,
0.01355230063199997,
0.062121737748384476,
0.05118407681584358,
0.036109987646341324,
0.07252847403287888,
-0.026156064122915268,
-0.1308245211839676,
-0.04783115163445473,
-0.056325800716876984,
0.03651336953043938,
-0.0032079205848276615,
0.06306126713752747,
0.006993431132286787,
-0.041772834956645966,
-0.0306546613574028,
0.01147886086255312,
-0.09781812876462936,
0.0014455860946327448,
0.017513660714030266,
0.03740357607603073,
0.032175563275814056,
0.029007544741034508,
0.031169990077614784,
-0.00591325294226408,
-0.09638972580432892,
-0.06541739404201508,
0.04041397199034691,
6.798556213573959e-32,
-0.039453715085983276,
-0.02702164836227894,
0.0008923521381802857,
0.022035473957657814,
0.000701828976161778,
0.009579380974173546,
0.0682465210556984,
-0.03845962509512901,
0.00012972287368029356,
-0.010259309783577919,
0.06606831401586533,
0.017696157097816467,
-0.043852753937244415,
-0.022540666162967682,
0.05737088993191719,
0.04299945384263992,
0.02516152523458004,
-0.06722787767648697,
-0.03930407389998436,
0.03383120521903038,
-0.003742788452655077,
0.030713943764567375,
-0.11785609275102615,
0.041776884347200394,
0.051059696823358536,
-0.04320888966321945,
-0.011751537211239338,
0.05608668550848961,
-0.0015987748047336936,
-0.0855453833937645,
-0.024703022092580795,
0.06486683338880539,
-0.02423456683754921,
-0.03812965378165245,
0.03875737264752388,
0.06930766254663467,
-0.027261054143309593,
-0.022288255393505096,
-0.05945197492837906,
0.0036702859215438366,
-0.02438408136367798,
-0.06186412274837494,
-0.04717127978801727,
0.04669870808720589,
-0.06258998066186905,
0.013387417420744896,
0.017214195802807808,
-0.023473963141441345,
0.013648184016346931,
-0.028203940019011497,
-0.052984025329351425,
0.07565651088953018,
0.001977774081751704,
-0.053817033767700195,
0.019487882032990456,
0.02443244494497776,
0.025735823437571526,
-0.05738282576203346,
0.05081719532608986,
0.006638266611844301,
-0.08367951214313507,
0.10216883569955826,
0.0862242802977562,
-0.04985545948147774
] |
26
Chapter 2
c (x + 3)2 > 0
So the minimum value of f( x) is β14.
This occurs when ( x + 3)2 = 0,
so when x = β 3A squared value must be greater than or equal to 0.
Find the roots of the function f(x) = x6 + 7x3 β 8, xΒ β β .Example 10
f(x) = 0
x6 + 7x3 β 8 = 0
(x3)2 + 7( x3) β 8 = 0
(x3 β 1)( x3 + 8) = 0
So x3 = 1 or x3 = β8
x3 = 1 β x = 1
x3 = β8 β x = β 2
The roots of f( x) are 1 and β 2.
Alternatively, let u = x3.
f(x) = x6 + 7x3 β 8
= (x3)2 + 7( x3) β 8
= u2 + 7u β 8
= (u β 1)( u + 8)
So when f( x) = 0, u = 1 or u = β 8.
If u = 1 β x3 = 1 β x = 1
If u = β 8 β x3 = β8 β x = β 2
The roots of f( x) are 1 and β 2.Treat x3 as a single variable and factorise.
Solve the quadratic equation to find two values
for x3, then find the corresponding values of x.
You can simplify this working with a substitution.
Replace x3 with u and solve the quadratic
equation in u.
The s olutions to the quadratic
equation will be values of u. Convert back to values of x using your substitution.Watch out
1 Using the functions f(x)
= 5x + 3, g(x) = x2 β 2 and h(x) = ββ―_____ x + 1 , find the values of:
a f(1) b g(3) c h(8) d f(1.5) e g ( ββ―__
2 )
f h (β1) g f(4) + g(2) h f(0) + g(0)
+ h(0) i g(4) ____ h(3)
2 The function f(x) is defined b
y f(x) = x2 β 2x, x β β .
Giv
en that f(a) = 8, find two possible values for a.
3 Find all of the r
oots of the following functions:
a f(x)
= 10 β 15x b g(x)
= (x + 9)(x β 2) c h(x)
= x2 + 6x β 40
d j(x)
= 144 β x2 e k(x) = x(x + 5)(x + 7) f m(x) = x3 + 5x2 β 24xSubstitute x = a into the function and
set the resulting expression equal to 8.Problem-solving PExercise 2Ef(x) can be written as a function of a function.
The only powers of x in f( x) are 6, 3 and 0 so you
can write it as a quadratic function of x3.Problem-solving(x + 3)2 > 0 so (x + 3)2 β 14 > β14
|
[
0.00841620471328497,
0.1422567218542099,
-0.006873862352222204,
-0.048969000577926636,
0.04064157232642174,
0.045137301087379456,
-0.08182214200496674,
0.02331303432583809,
-0.057616159319877625,
0.08946049213409424,
0.03645595163106918,
-0.06614738702774048,
-0.01590130478143692,
-0.0050530279986560345,
0.026708392426371574,
0.016262372955679893,
-0.06396257132291794,
0.027727050706744194,
-0.08377405256032944,
-0.0424375981092453,
-0.05823790282011032,
-0.06054249405860901,
-0.019325915724039078,
-0.07147281616926193,
0.08991996198892593,
-0.064756378531456,
-0.017942527309060097,
-0.09858014434576035,
0.01608992926776409,
0.03203396871685982,
-0.0669044628739357,
-0.017577122896909714,
0.04843069240450859,
-0.051607608795166016,
0.10608801990747452,
-0.0036558746360242367,
0.014594866894185543,
0.00687397550791502,
0.031667307019233704,
0.00823515560477972,
-0.06991780549287796,
0.043670687824487686,
0.005838209763169289,
-0.03452247381210327,
0.039128679782152176,
-0.006021654233336449,
-0.049330875277519226,
-0.030539749190211296,
-0.00371786137111485,
0.006108790170401335,
0.01834297925233841,
0.1007368266582489,
0.02334795705974102,
0.08387750387191772,
-0.030885081738233566,
-0.03538333624601364,
0.01978204771876335,
-0.007662774529308081,
0.027587946504354477,
0.04081020876765251,
0.003921096213161945,
0.08153475821018219,
-0.02116629295051098,
0.0460149347782135,
0.05024438351392746,
0.0842304676771164,
-0.018095605075359344,
-0.027482692152261734,
-0.006166248116642237,
0.06976337730884552,
-0.05540524795651436,
0.05562210828065872,
-0.013747459277510643,
0.01672343537211418,
-0.017469104379415512,
0.004365073051303625,
0.026372410356998444,
-0.06441224366426468,
0.03543587028980255,
0.013751867227256298,
0.02509959228336811,
0.007889355532824993,
0.05310710892081261,
0.12348566204309464,
0.007657253183424473,
0.009502683766186237,
0.14751873910427094,
0.0029644332826137543,
-0.025886882096529007,
0.016271188855171204,
-0.0038610503543168306,
0.043039049953222275,
-0.027992144227027893,
-0.020831560716032982,
-0.029184089973568916,
-0.11878158152103424,
0.0016163187101483345,
-0.09690084308385849,
0.0259955246001482,
0.053950756788253784,
-0.04397932440042496,
-0.01885714940726757,
0.00379078253172338,
0.040591850876808167,
-0.02470390312373638,
-0.05800448730587959,
0.034238167107105255,
-0.0452294647693634,
0.04431057721376419,
-0.11222074180841446,
-0.03817599266767502,
-0.12035819888114929,
-0.06624210625886917,
0.018337620422244072,
0.057849783450365067,
-0.001605688245035708,
0.01281167846173048,
0.0016635616775602102,
-0.03905533626675606,
-0.029369588941335678,
-0.02081226371228695,
-0.06846529990434647,
0.061934325844049454,
-0.06762968748807907,
-0.049271758645772934,
-0.040234800428152084,
0.015346333384513855,
0.022038521245121956,
0.03588789701461792,
-0.06382337957620621,
0.05494951829314232,
-0.09713605046272278,
-0.0500735267996788,
-0.01602335087954998,
-0.011871111579239368,
-0.03228253126144409,
-0.11108298599720001,
0.0661395788192749,
0.07757332175970078,
-0.044189684092998505,
0.019527943804860115,
-0.013417329639196396,
-0.029013516381382942,
-0.0121665233746171,
-0.016325008124113083,
-0.113665871322155,
-0.09201905131340027,
-0.037294358015060425,
-0.07532108575105667,
-0.03582246974110603,
-0.052938126027584076,
-0.040903136134147644,
-0.0012164274230599403,
0.08970827609300613,
0.009410425089299679,
-0.11959310621023178,
-0.04512694105505943,
-0.037481773644685745,
0.041364457458257675,
-0.04884025454521179,
-0.05866998806595802,
0.05366991460323334,
0.014579174108803272,
0.052923236042261124,
0.06647179275751114,
0.018413754180073738,
0.06260782480239868,
-0.02788436971604824,
-0.03688674792647362,
-0.028617609292268753,
0.023021748289465904,
0.02976696752011776,
0.047301020473241806,
0.057367999106645584,
0.02957506664097309,
0.013501035049557686,
0.059016887098550797,
0.0451258048415184,
0.03141983225941658,
0.1134931743144989,
0.0937691256403923,
0.059790968894958496,
0.0014347879914566875,
-0.023371728137135506,
-0.012870424427092075,
0.0542636439204216,
0.04297054931521416,
-0.00742369145154953,
0.021764323115348816,
-0.0507727786898613,
0.001294187968596816,
-0.031999845057725906,
-0.0029713583644479513,
0.054182566702365875,
-0.1371113359928131,
-0.03483991697430611,
0.04370933398604393,
-0.0022636738140136003,
-0.10458202660083771,
0.06435886770486832,
0.048160724341869354,
-0.06101255118846893,
0.09149771183729172,
0.033720728009939194,
-0.0232999250292778,
0.002509069861844182,
-0.11883126199245453,
-0.018612327054142952,
0.006254477426409721,
-0.014180129393935204,
-0.07215059548616409,
-0.010414615273475647,
0.006382114719599485,
-0.051047731190919876,
-0.025636723265051842,
-0.024901216849684715,
0.06990501284599304,
-0.043469782918691635,
-0.13613514602184296,
0.009691147133708,
-0.07479244470596313,
-0.026110181584954262,
0.0851837769150734,
-4.523607155811407e-33,
-0.035786569118499756,
-0.02265951782464981,
-0.03966544568538666,
-0.007652148138731718,
0.0091734379529953,
-0.02769288420677185,
0.025324521586298943,
-0.030366364866495132,
0.031160345301032066,
0.06945084035396576,
0.03836006671190262,
0.01514457631856203,
-0.03823128715157509,
-0.02174873650074005,
-0.035151299089193344,
0.005919911433011293,
-0.09006598591804504,
-0.02198595553636551,
0.03193320333957672,
0.07556047290563583,
0.03175082430243492,
0.0472605936229229,
-0.021479852497577667,
0.00987942237406969,
0.03446469083428383,
0.0039041920099407434,
0.0897672176361084,
-0.04504326730966568,
0.005394496023654938,
0.023139864206314087,
-0.03844936564564705,
0.02998019941151142,
0.04546871408820152,
0.09193283319473267,
-0.0054927091114223,
-0.04387067258358002,
0.003271728754043579,
-0.055932629853487015,
0.012503924779593945,
-0.025980036705732346,
0.1071915477514267,
0.09584686905145645,
0.05830619856715202,
-0.0101079773157835,
0.020227646455168724,
-0.002646992215886712,
0.040772873908281326,
0.005671541206538677,
-0.015480446629226208,
0.0643562600016594,
0.029280126094818115,
-0.048953961580991745,
0.03166439011693001,
0.06288348883390427,
0.021648768335580826,
-0.0071684508584439754,
-0.05527128279209137,
-0.010324465110898018,
0.012869169004261494,
0.01579618640244007,
-0.031167106702923775,
-0.06513962149620056,
-0.039238736033439636,
0.034426890313625336,
-0.07130897045135498,
0.015768665820360184,
-0.04092332348227501,
-0.013118892908096313,
0.02803960256278515,
-0.03147650882601738,
0.0010003719944506884,
-0.01824282854795456,
0.06714963167905807,
-0.053413283079862595,
-0.12031203508377075,
0.06468000262975693,
-0.04884548857808113,
0.019599927589297295,
0.062424272298812866,
-0.1516655534505844,
-0.04309253767132759,
0.04923709109425545,
-0.011817356571555138,
-0.05592042952775955,
-0.05847428739070892,
0.007615461014211178,
-0.0003696007188409567,
0.02215150184929371,
0.11969144642353058,
0.02819780446588993,
-0.018449485301971436,
0.041493721306324005,
-0.018405558541417122,
-0.0503179207444191,
0.0734543725848198,
7.351295991723264e-32,
-0.046632956713438034,
-0.002689030487090349,
-0.03716808557510376,
0.018349913880228996,
0.05746496096253395,
0.012391791678965092,
-0.0197033379226923,
-0.031026316806674004,
-0.023342804983258247,
0.02027427777647972,
0.0407240092754364,
0.05673280730843544,
-0.057629380375146866,
-0.0006286108400672674,
-0.0877232700586319,
-0.010912680067121983,
-0.0019487484823912382,
-0.03624590113759041,
-0.02390313521027565,
0.006945126224309206,
-0.039525579661130905,
0.029813354834914207,
-0.0727672353386879,
0.0590534545481205,
0.11224687844514847,
0.0018270313739776611,
-0.0029643666930496693,
0.005262105725705624,
0.045684922486543655,
0.020119942724704742,
-0.0015412281500175595,
0.0362384170293808,
-0.01596543937921524,
0.017468959093093872,
-0.0021079310681670904,
-0.0641578957438469,
0.03168873488903046,
-0.07480469346046448,
-0.05118998885154724,
-0.02665114961564541,
-0.028486842289566994,
-0.0011703985510393977,
-0.09357486665248871,
-0.04275970906019211,
-0.06786178797483444,
-0.03856239095330238,
-0.0486319400370121,
0.025844043120741844,
0.06880151480436325,
0.050377730280160904,
0.009126956574618816,
0.06891263276338577,
0.11646506935358047,
0.0355948805809021,
0.02625592052936554,
-0.04875783994793892,
-0.047732073813676834,
-0.03626824542880058,
-0.0031876382417976856,
-0.0044301655143499374,
-0.055511780083179474,
0.07636008411645889,
-0.02150210738182068,
0.06316734850406647
] |
27Quadratics
4 The functions p and q are giv
en by p(x) = x2 β 3x and q(x) = 2x β 6, x β β .
Find the two v
alues of x for which p(x) = q(x).
5 The functions f and g are gi
ven by f(x) = 2x3 + 30x and g(x) = 17x2, Β x β β .
Find the three v
alues of x for which f(x) = g(x).
6 The function f is defined as f(x
) = x2 β 2x + 2, x β β .
a Write f(x
) in the form (x + p)2 + q, where p and q are constants to be found. (2 marks)
b Hence, or otherwise, e
xplain why f(x) > 0 for all values of x, and find the minimum
value of f(x). (1 mark)
7 Find all roots of
the following functions:
a f(x)
= x6 + 9x3 + 8 b g(x) = x4 β 12x2 + 32
c h(x)
= 27x6 + 26x3 β 1 d j(x) = 32x10 β 33x5 + 1
e k(x)
= x β 7 ββ―__
x + 10 f m(x) = 2 x 2 _ 3 + 2 x 1 _ 3 β 12
8 The function f is defined as f(x
) = 32x β 28(3x) + 27, x β β .
a Write f(x
) in the form (3x β a)(3x β b), where a and b are
real constants. (2 marks)
b Hence find the two roots of
f(x). (2 marks)E
The function in
par
t b has four roots.Hint
E/P
Consider f( x) as a
function of a function.Problem-solving
2.4 Quadratic graphs
When f(x) = ax2 + bx + c, the graph of y = f(x) has a curved shape called a parabola.
You can sketch a quadratic graph by identifying key features.
The coefficient of x2 determines the overall shape of the graph.
When a is positive the parabola will have this shape:
When a is negative the parabola will have this shape:
y
x Oy
x O1 The graph crosses the y-axis when
x
= 0. The y-coordinate is equal to c.
3 Quadratic graphs have one turning point. This can be a minimum or a maximum. Sinc
eΒ a parabola is symmetrical, the turning
point and line of symmetry are half-way between the two roots. 2
The graph crosses the x-axis when y = 0. The x-c
oordinates are roots of the function f(x). 11
2 2 2 2
33
β You can find the coordinates of the turning point
of a quadr
atic graph by completing the square.
If f(x) = a(x + p)2 + q, the graph of y = f(x) has a
turning point at (β p, q). The graph of y = a (x + p )2 + q
is a translation of the graph of
y = ax2 by ( βp q ) . β Section 4.5Links
|
[
0.010665295645594597,
0.0990675538778305,
-0.0009615758899599314,
0.0062354570254683495,
-0.055398549884557724,
0.11538945138454437,
0.014974798075854778,
-0.0245650764554739,
-0.03978198394179344,
0.008517695590853691,
0.00008024739508982748,
-0.05250926315784454,
-0.02854674495756626,
-0.058282751590013504,
0.08243124186992645,
-0.04158555716276169,
-0.039552073925733566,
0.001369082136079669,
-0.09440642595291138,
0.022613124921917915,
0.034985896199941635,
-0.04436487704515457,
-0.07825424522161484,
-0.1135011613368988,
0.10786943882703781,
-0.00026015398907475173,
0.016512351110577583,
0.11350030452013016,
-0.006351151969283819,
-0.024720337241888046,
0.026765184476971626,
0.02079702727496624,
0.03452384099364281,
-0.07394883036613464,
0.06069294735789299,
0.046354737132787704,
-0.05145507678389549,
0.028139635920524597,
0.08088421821594238,
-0.010220503434538841,
-0.037023380398750305,
0.00928258802741766,
0.00946350209414959,
-0.06923943012952805,
0.07262510806322098,
-0.0198034830391407,
0.023585688322782516,
0.013859536498785019,
0.036416541785001755,
0.00378375593572855,
0.00630585104227066,
0.08720226585865021,
-0.06618218123912811,
0.09131158143281937,
0.03783633932471275,
-0.0797698125243187,
0.04266257956624031,
0.012229422107338905,
0.031230606138706207,
0.030348442494869232,
-0.06479942053556442,
0.049441516399383545,
0.03168211132287979,
0.10223155468702316,
-0.02067355066537857,
0.010164482519030571,
-0.0721009373664856,
-0.0481606125831604,
-0.0029190450441092253,
0.03251511603593826,
-0.05578707531094551,
0.0874028354883194,
-0.05615933984518051,
-0.04093780741095543,
-0.040201544761657715,
0.009801149368286133,
-0.019770033657550812,
-0.014039627276360989,
-0.024672525003552437,
0.017034294083714485,
0.027822785079479218,
-0.006571450270712376,
0.05734192952513695,
0.09185615181922913,
-0.027426421642303467,
0.04166237264871597,
0.047626521438360214,
0.03779743239283562,
0.044458240270614624,
-0.00021126429783180356,
-0.012174921110272408,
0.02687344141304493,
-0.002431647852063179,
-0.08630852401256561,
0.03801177069544792,
-0.061773426830768585,
0.03211282566189766,
-0.03564971312880516,
0.021365342661738396,
0.0861712396144867,
0.0391542911529541,
0.026254979893565178,
0.04792044311761856,
0.0844026654958725,
-0.015322008170187473,
0.02183588035404682,
0.04261334612965584,
-0.07838787883520126,
0.015844734385609627,
-0.01799003966152668,
0.006666970904916525,
-0.02832985110580921,
-0.010978873819112778,
0.031408075243234634,
0.10548709332942963,
-0.048851363360881805,
-0.003601467004045844,
0.01361805573105812,
0.021808451041579247,
-0.06972522288560867,
0.0346047580242157,
-0.10804061591625214,
0.041238583624362946,
-0.033112168312072754,
-0.008855171501636505,
-0.05333034321665764,
0.025438988581299782,
-0.0011065982980653644,
-0.06573138386011124,
-0.042529571801424026,
0.023887841030955315,
-0.038122765719890594,
-0.029559781774878502,
-0.060789406299591064,
0.0112835131585598,
0.014232492074370384,
-0.1357189416885376,
0.02869207039475441,
-0.025239596143364906,
0.02761371061205864,
0.03792084753513336,
0.08485084772109985,
0.06189078092575073,
0.005554371979087591,
0.05003936216235161,
0.01442298386245966,
-0.05748842656612396,
-0.022460568696260452,
-0.039967283606529236,
-0.022815456613898277,
-0.001558785093948245,
-0.021809615194797516,
0.021502196788787842,
0.1013227328658104,
0.013867650181055069,
-0.060606442391872406,
-0.07088947296142578,
-0.02363373339176178,
-0.009001984260976315,
-0.04018598794937134,
-0.08624628186225891,
0.01787496916949749,
-0.08208061754703522,
0.099190853536129,
0.02415647730231285,
0.08228699117898941,
0.05164829269051552,
0.03582671284675598,
-0.013032726012170315,
-0.01813516393303871,
0.011238240636885166,
0.00726698525249958,
-0.01856897957623005,
-0.017259923741221428,
0.005341488867998123,
-0.010256893001496792,
0.05711090937256813,
0.01375493686646223,
-0.0368703193962574,
0.06489672511816025,
0.07701239734888077,
-0.009716549888253212,
-0.007824626751244068,
-0.0516546294093132,
0.02521614357829094,
0.004296422004699707,
-0.0473870225250721,
0.01366458460688591,
-0.002243937226012349,
-0.014114194549620152,
0.04415527358651161,
-0.0810776948928833,
0.03468942642211914,
0.03342472389340401,
-0.11606351286172867,
0.007192512974143028,
-0.029230447486042976,
0.0836334079504013,
-0.0545651912689209,
0.028288746252655983,
0.08236140757799149,
-0.016054300591349602,
0.07373987883329391,
-0.06071953475475311,
-0.086883544921875,
-0.04406490549445152,
-0.1006409227848053,
-0.016857478767633438,
-0.039073437452316284,
0.02913513034582138,
-0.0706799253821373,
0.036939121782779694,
-0.05867701396346092,
-0.06247550621628761,
0.055060192942619324,
-0.0210503488779068,
0.005458883009850979,
-0.06562851369380951,
-0.14511217176914215,
0.002519579604268074,
-0.04160851240158081,
-0.03408817574381828,
0.05816138535737991,
-3.0152219883858633e-33,
-0.05334964394569397,
-0.0038471450097858906,
-0.0829220861196518,
0.0022171013988554478,
0.013642101548612118,
-0.001529291970655322,
0.015029223635792732,
-0.0936756581068039,
0.08071433007717133,
0.09458185732364655,
0.08302043378353119,
0.01576993800699711,
-0.0751512199640274,
0.056091345846652985,
0.023596879094839096,
0.013766007497906685,
-0.0329173319041729,
-0.023398656398057938,
-0.07577616721391678,
-0.021452771499753,
0.0951775535941124,
0.0492141954600811,
-0.05934158340096474,
-0.02838255651295185,
-0.01568402349948883,
0.052261050790548325,
0.08123499900102615,
-0.07363999634981155,
0.039264362305402756,
0.08011974394321442,
-0.015063634142279625,
0.03067472018301487,
0.0609915591776371,
0.06638213992118835,
-0.02137700282037258,
-0.031760405749082565,
0.06457824259996414,
0.03106161206960678,
-0.04028357192873955,
-0.05402417853474617,
0.09939854592084885,
0.033453866839408875,
-0.004531078971922398,
-0.002863198285922408,
0.05388690158724785,
0.007935832254588604,
0.04845369607210159,
0.026847263798117638,
-0.009306825697422028,
0.05252887308597565,
-0.049487922340631485,
-0.08679934591054916,
-0.018971145153045654,
0.020349806174635887,
0.06719236820936203,
-0.004890242591500282,
0.04533848911523819,
-0.06037706136703491,
0.061088595539331436,
-0.04130854830145836,
-0.04351939260959625,
-0.03375593572854996,
0.02214409038424492,
0.04965904727578163,
-0.09893254935741425,
-0.007742017041891813,
-0.004820769187062979,
-0.0030167843215167522,
0.03754011541604996,
0.0069734794087708,
0.032513003796339035,
-0.03261955454945564,
0.016080409288406372,
-0.10693500190973282,
-0.11783306300640106,
0.07977746427059174,
-0.07041045278310776,
0.060587070882320404,
0.012858342379331589,
-0.02846365235745907,
0.019291214644908905,
0.0065917205065488815,
-0.0006584660732187331,
0.02918967790901661,
-0.1486702263355255,
0.045222796499729156,
0.0375649556517601,
0.03082769550383091,
0.0798281654715538,
-0.04080758988857269,
-0.056905850768089294,
0.08867116272449493,
-0.0243659820407629,
-0.032838236540555954,
0.01311696507036686,
7.029907607747995e-32,
-0.047294240444898605,
-0.02767203189432621,
-0.06849642097949982,
-0.01363386306911707,
0.016279807314276695,
-0.03769936040043831,
-0.012237467803061008,
-0.0397755466401577,
0.016875585541129112,
-0.010090958327054977,
0.024795180186629295,
0.03238433972001076,
-0.06401239335536957,
-0.03146053105592728,
0.023563144728541374,
-0.03796663135290146,
-0.029944945126771927,
0.019290218129754066,
-0.004006647504866123,
-0.0477968230843544,
-0.06332182884216309,
0.029210438951849937,
-0.09998588263988495,
0.06219801679253578,
-0.003874291433021426,
0.01766868866980076,
-0.03936074301600456,
-0.05330738425254822,
-0.027803482487797737,
-0.047011494636535645,
0.040802787989377975,
0.04655393585562706,
-0.06679316610097885,
-0.06516572833061218,
0.04235917329788208,
0.037757255136966705,
-0.021976446732878685,
-0.008957858197391033,
0.014481323771178722,
0.012424103915691376,
-0.05149058252573013,
-0.07947277277708054,
-0.010451067239046097,
0.0057595293037593365,
-0.050763797014951706,
-0.023254796862602234,
0.04429289326071739,
-0.08124254643917084,
0.06408300250768661,
-0.07271097600460052,
-0.04076169431209564,
0.017413051798939705,
-0.029643477872014046,
-0.06703359633684158,
-0.024423236027359962,
0.0074807885102927685,
-0.029767949134111404,
0.07932029664516449,
0.03634646162390709,
-0.05209541693329811,
-0.09344704449176788,
0.12264671176671982,
-0.009097267873585224,
-0.048795368522405624
] |
28
Chapter 2
As a = 1 is positive, the graph has a
shape and a minimum point.
When x = 0, y = 4, so the graph crosses
the y-axis at (0, 4).
When y = 0,
x2 β 5 x + 4 = 0
(x β 1)( x β 4) = 0
x = 1 or x = 4, so the graph crosses the x-axis at (1, 0) and (4, 0).
Completing the square:
x
2 β 5 x + 4 = (x β 5 _ 2 ) 2 β 25 __ 4 + 4
= (x β 5 _ 2 ) 2 β 9 _ 4
So th
e minimum point has coordinates
( 5 _ 2 , β 9 _ 4 ) .
Alternatively, the minimum occurs when
x is half-way between 1 and 4,
so x = 1 + 4 _______ 2 = 5 _ 2
y = ( 5 _ 2 ) 2 β 5 Γ ( 5 _ 2 ) + 4 = β 9 _ 4
so th
e minimum has coordinates ( 5 _ 2 , β 9 _ 4 ) .
The sketch of the graph is:
O xy
/four.ss01
1 /four.ss01
, β5
294()Use the coefficient of x2 to determine the general
shape of the graph.
This example factorises, but you may need to use
the quadratic formula or complete the square.
Complete the square to find the coordinates of the turning point.
You could use a graphic calculator or substitute some values to check your sketch.
When x = 5, y = 5
2 β 5 Γ 5 + 4 = 4.Sketch the graph of y = x2 β 5x + 4, and find the coordinates of its turning point.Example 11
If yo u use symmetry to find the
x-coordinate of the minimum point, you need to
substitute this value into the equation to find the
y-coordinate of the minimum point.Watch out
Explore how the graph of
y
= (x + p )2 + q changes as the values of p
and q change using GeoGebra.Online
|
[
0.03143851459026337,
0.0633617490530014,
-0.02836621180176735,
-0.09260756522417068,
-0.03586795553565025,
0.06915653496980667,
-0.0033093818929046392,
0.0765618309378624,
-0.07633975893259048,
0.07838607579469681,
0.07753565907478333,
-0.06027836352586746,
0.00949497427791357,
0.04015395790338516,
-0.02700657770037651,
0.029212797060608864,
-0.0514465868473053,
0.007270483765751123,
-0.007627070881426334,
-0.05983876436948776,
0.018576206639409065,
-0.0764966830611229,
-0.00725672347471118,
-0.06584181636571884,
0.04581683874130249,
-0.10818992555141449,
0.0214841291308403,
-0.09069361537694931,
-0.019599078223109245,
-0.03520045056939125,
0.012313131242990494,
-0.02898765727877617,
0.01506554801017046,
-0.010888345539569855,
0.038438644260168076,
0.01124107837677002,
0.06469232589006424,
0.016773154959082603,
0.078895702958107,
0.00979881826788187,
-0.008907235227525234,
0.019512677565217018,
-0.08075340837240219,
0.003467704402282834,
-0.012740052305161953,
-0.032981518656015396,
-0.042543523013591766,
-0.05597629398107529,
0.07171551138162613,
-0.02978605404496193,
0.023510608822107315,
0.010871362872421741,
0.014409665949642658,
0.02634553797543049,
-0.058381181210279465,
0.015131481923162937,
0.01483877468854189,
-0.01243260595947504,
0.004030903801321983,
0.11799154430627823,
0.06923341751098633,
0.0027673672884702682,
-0.07766386866569519,
0.0479850210249424,
0.03539334982633591,
0.03490397706627846,
0.08220785111188889,
-0.039491090923547745,
-0.00003313180786790326,
0.06830842047929764,
-0.09340637922286987,
-0.05053718388080597,
0.013797776773571968,
-0.07560370117425919,
0.022771663963794708,
-0.010586747899651527,
0.021807268261909485,
0.03467787057161331,
0.02463757060468197,
-0.04052223265171051,
-0.046062298119068146,
0.050043847411870956,
0.06087872385978699,
0.03874947130680084,
-0.002074739895761013,
-0.0198445376008749,
0.07200776785612106,
-0.0382583886384964,
-0.02600683830678463,
-0.0014629432698711753,
0.03978459909558296,
0.004183493554592133,
-0.07109653204679489,
-0.018920961767435074,
0.03319101780653,
-0.19309896230697632,
0.0719536766409874,
-0.05463078245520592,
0.0023750897962599993,
0.10106833279132843,
-0.04860461503267288,
0.05319838225841522,
0.046108976006507874,
0.02645621821284294,
-0.039282724261283875,
0.02694152109324932,
0.0787321999669075,
-0.04325859993696213,
-0.02412310615181923,
-0.09477660804986954,
-0.016111580654978752,
-0.08710932731628418,
0.03358937427401543,
0.11020559072494507,
0.08478370308876038,
-0.04896582290530205,
0.04066600650548935,
-0.04543057456612587,
-0.004153755027800798,
-0.07764890789985657,
0.06848760694265366,
-0.036776233464479446,
0.0029476864729076624,
-0.023919282481074333,
-0.04509347677230835,
-0.008646998554468155,
-0.028689032420516014,
0.040052663534879684,
0.013006621971726418,
-0.09826532751321793,
0.06887249648571014,
-0.09982959181070328,
-0.021924346685409546,
-0.031920045614242554,
-0.047661762684583664,
-0.04576684162020683,
-0.040025003254413605,
0.10626058280467987,
-0.07687077671289444,
-0.08550934493541718,
-0.09034457802772522,
0.04813944175839424,
0.03605194762349129,
-0.028106529265642166,
0.01736041158437729,
-0.022523412480950356,
-0.028793003410100937,
-0.010493047535419464,
0.023391500115394592,
0.044307783246040344,
0.02324763871729374,
-0.03435225784778595,
0.006783179473131895,
0.07695650309324265,
0.01489103864878416,
-0.040679242461919785,
0.008027277886867523,
0.019756415858864784,
0.031209494918584824,
-0.05895690247416496,
-0.04361819848418236,
0.01152061764150858,
-0.011200439184904099,
-0.0006700627272948623,
0.012366979382932186,
0.0217796191573143,
-0.007788476534187794,
0.026687223464250565,
0.07922159135341644,
-0.02427198551595211,
0.005873309448361397,
-0.03640676289796829,
-0.015613396652042866,
0.09212274849414825,
-0.010931069031357765,
0.04488804191350937,
0.09923700243234634,
0.0905739888548851,
-0.003215686185285449,
0.046203404664993286,
0.02399132028222084,
-0.05817472189664841,
0.00042684373329393566,
-0.0022356745321303606,
0.05783320963382721,
-0.03296148031949997,
-0.034816496074199677,
-0.029791323468089104,
0.019773690029978752,
-0.0037613497115671635,
0.010963181033730507,
-0.015508209355175495,
0.0030585136264562607,
0.07703584432601929,
-0.15763626992702484,
-0.029437458142638206,
0.08730559796094894,
0.004400565288960934,
-0.0664738118648529,
0.04170835018157959,
0.039568137377500534,
-0.034626442939043045,
0.09629058837890625,
0.05296444520354271,
-0.004464165307581425,
-0.058750078082084656,
-0.006669104564934969,
-0.07347467541694641,
-0.0630556046962738,
0.019284730777144432,
-0.009143976494669914,
-0.010017892345786095,
-0.06644156575202942,
-0.05091208964586258,
0.020137624815106392,
-0.0252617709338665,
0.049328941851854324,
-0.042369645088911057,
-0.10796793550252914,
-0.05810996890068054,
-0.05585765838623047,
0.06425826996564865,
0.0027650166302919388,
4.202822094311697e-33,
-0.001965971663594246,
-0.002025587484240532,
-0.03592800348997116,
-0.06964574009180069,
-0.00400262838229537,
-0.04046538472175598,
0.07765085250139236,
-0.023315835744142532,
0.03464611992239952,
0.16941846907138824,
-0.02845669724047184,
0.011407570913434029,
-0.03717939555644989,
0.0307648703455925,
-0.03316311538219452,
0.017676811665296555,
0.011269591748714447,
0.04382818937301636,
-0.004498278722167015,
-0.07312364131212234,
0.060121309012174606,
0.020248573273420334,
0.024585267528891563,
-0.027111362665891647,
0.06278865039348602,
0.05012359470129013,
0.05113450065255165,
-0.043432123959064484,
-0.004228026606142521,
0.017475804314017296,
-0.06580964475870132,
0.023984946310520172,
0.01459216233342886,
0.021526772528886795,
0.03908809274435043,
-0.0038481643423438072,
0.00950479507446289,
-0.10086261481046677,
0.02188855968415737,
-0.016590185463428497,
0.026909682899713516,
0.010569295845925808,
0.0873490646481514,
0.030566586181521416,
0.018871065229177475,
0.07653047889471054,
0.04653814807534218,
0.05023420229554176,
0.0014546883758157492,
-0.00687152286991477,
-0.028888309374451637,
0.018563803285360336,
0.13123314082622528,
-0.01568698324263096,
0.03556673973798752,
-0.014252937398850918,
-0.010923923924565315,
-0.03272515907883644,
-0.007407107390463352,
0.021573740988969803,
-0.010815909132361412,
-0.05905533954501152,
-0.034202076494693756,
0.05403531715273857,
-0.008430957794189453,
-0.04886145517230034,
0.021592820063233376,
-0.03673773258924484,
0.06862663477659225,
0.01745530404150486,
-0.07390926778316498,
0.04452834650874138,
0.008455662988126278,
-0.051744621247053146,
-0.05180865526199341,
-0.04503229260444641,
-0.0639721229672432,
0.015465275384485722,
0.07400064915418625,
-0.055713485926389694,
-0.12750989198684692,
0.06925739347934723,
-0.024903293699026108,
-0.046306315809488297,
-0.003847050480544567,
-0.012735109776258469,
0.036637965589761734,
0.03323494270443916,
0.1381952315568924,
0.03757888451218605,
0.0018427520990371704,
-0.04712368920445442,
-0.06232481449842453,
0.034332387149333954,
-0.013294233940541744,
6.851498128146293e-32,
-0.028727786615490913,
-0.02633601427078247,
-0.012091840617358685,
0.021145209670066833,
0.02710849419236183,
0.030176065862178802,
0.06525111198425293,
-0.009460094384849072,
-0.02602141909301281,
-0.012360048480331898,
0.05044619366526604,
0.07681173086166382,
-0.11956287920475006,
0.08424734324216843,
-0.09834276884794235,
-0.04484114795923233,
-0.03718443959951401,
-0.06664517521858215,
-0.06461472064256668,
-0.022033007815480232,
0.01117949839681387,
-0.049364157021045685,
-0.10203222185373306,
-0.04169852286577225,
0.0035909328144043684,
0.02813614532351494,
-0.05837531387805939,
-0.018702547997236252,
0.05793043226003647,
-0.02732187695801258,
-0.020431077107787132,
-0.013345817103981972,
-0.004490492399781942,
-0.002283351728692651,
-0.013674602843821049,
-0.06665073335170746,
-0.020200438797473907,
0.05990254878997803,
-0.022496772930026054,
-0.02013048715889454,
-0.0403580479323864,
-0.0356031097471714,
0.00007140928209992126,
0.0011749223340302706,
-0.034238170832395554,
0.008091957308351994,
0.053771715611219406,
-0.02314620278775692,
-0.007186868693679571,
0.015071657486259937,
0.028654415160417557,
0.07416696101427078,
0.11965636163949966,
0.0875590592622757,
0.038386519998311996,
-0.07821027934551239,
-0.009003604762256145,
-0.013845750130712986,
-0.021714283153414726,
0.003928611520677805,
-0.06231561675667763,
0.11279404908418655,
-0.06832746416330338,
0.031548164784908295
] |
29Quadratics
As a = β 2 is negative, the graph has a
shape and a maximum point.
When x = 0, y = β 3, so the graph
crosses the y -axis at (0, β 3).
When y = 0,β2x
2 + 4x β 3 = 0
Using the quadratic formula,
x = β4 Β± β _____________ 4 2 β 4 (β2) (β3) ____________________ 2 Γ (β2)
x =
β4 Β± β ____ β8 __________ β4
The
re are no real solutions, so the graph
does not cross the x -axis.
Completing the square:
β2x2 + 4x β 3
= β2(x2 β 2x) β 3
= β2((x β 1)2 β 1) β 3
= β2(x β 1)2 + 2 β 3
= β2(x β 1)2 β 1
So the maximum point has coordinates
(1, β1).
The line of symmetry is vertical and goes through the maximum point. It has the equation x = 1.
O xy
(1, β1)
β3a = β2, b = 4 and c = β3Sketch the graph of y = 4x β 2x2 β 3. Find the coordinates of its turning point and write down the
equation of its line of symmetry.Example 12
A ske tch graph does not need to be
plotted exactly or drawn to scale. However you
should:β
dra
w a smooth curve by hand
β ide
ntify any relevant key points (such as
intercepts and turning points)
β lab
el your axes.Watch outItβs easier to see that a , 0 if you write the equation in the form y = β2x
2 + 4x β 3.
You would need to square root a negative number to evaluate this expression. Therefore this equation has no real solutions.
The coefficient of x2 is β2 so take out a factor of β 2
|
[
0.08585786819458008,
0.020747842267155647,
-0.005412574391812086,
-0.05917303264141083,
-0.07698357850313187,
0.029789339751005173,
0.004894026089459658,
0.055436842143535614,
-0.1274033486843109,
0.03615638613700867,
0.05059301108121872,
-0.07606087625026703,
-0.012585122138261795,
0.02288830280303955,
0.011432381346821785,
0.020809881389141083,
-0.03129007667303085,
0.07209262996912003,
-0.04158734530210495,
-0.005355534143745899,
0.030597170814871788,
-0.1388261467218399,
0.044918596744537354,
-0.09341809153556824,
0.09477990865707397,
-0.0574367381632328,
-0.0008528124890290201,
0.012558931484818459,
-0.0227789506316185,
0.06691592186689377,
-0.04265032708644867,
-0.030511097982525826,
-0.007490914314985275,
-0.07505976408720016,
0.07959384471178055,
0.00222486793063581,
0.039788320660591125,
0.01861804910004139,
0.030230358242988586,
-0.03680678829550743,
-0.010759076103568077,
0.020311323925852776,
-0.00937694776803255,
-0.03871816396713257,
0.06413687765598297,
-0.137508824467659,
-0.019662579521536827,
-0.03121340274810791,
0.12526708841323853,
0.0038306654896587133,
0.07528427243232727,
0.11016856878995895,
-0.009860447607934475,
0.08994139730930328,
-0.024011103436350822,
-0.010308068245649338,
-0.013559387996792793,
-0.011642780154943466,
-0.021639801561832428,
0.0055405921302735806,
0.06405486166477203,
0.05638720467686653,
0.004903736524283886,
0.06757217645645142,
0.024981413036584854,
0.006634566001594067,
-0.029837455600500107,
-0.06582508981227875,
-0.0010827871738001704,
0.05969277396798134,
0.027644317597150803,
-0.018981212750077248,
0.017402097582817078,
0.010248093865811825,
-0.032326839864254,
-0.0042231460101902485,
0.018573755398392677,
0.06185419112443924,
-0.016237959265708923,
-0.02672261744737625,
-0.02487206645309925,
0.04315272718667984,
0.11516974121332169,
0.035347919911146164,
-0.0011713763233274221,
0.002261156914755702,
0.016788577660918236,
-0.05445818230509758,
0.007718772627413273,
0.10243338346481323,
-0.005787145346403122,
0.02552889660000801,
-0.11241376399993896,
-0.03014751337468624,
-0.005108600482344627,
-0.09795165061950684,
0.06382986903190613,
-0.04684579372406006,
0.009684276767075062,
0.07628300786018372,
-0.07545196264982224,
-0.009249802678823471,
-0.015700774267315865,
0.04580438882112503,
0.00989600270986557,
-0.06159243360161781,
-0.013008812442421913,
0.0008435393683612347,
-0.009352446533739567,
-0.06535901129245758,
-0.05215798318386078,
-0.04439923167228699,
0.028162628412246704,
0.04777314141392708,
0.08543693274259567,
-0.08297612518072128,
-0.007970493286848068,
-0.03570833429694176,
0.01760711520910263,
-0.017713962122797966,
0.09725674986839294,
0.037090450525283813,
0.0388517864048481,
-0.0519217923283577,
0.006080738268792629,
0.1191626489162445,
-0.03394903242588043,
0.061905793845653534,
-0.0319758765399456,
-0.08273212611675262,
0.050686389207839966,
-0.0968867763876915,
-0.04420400410890579,
-0.06571809947490692,
-0.043919164687395096,
-0.024017857387661934,
-0.06868261098861694,
0.11548040807247162,
0.037413883954286575,
-0.045619748532772064,
-0.03283126279711723,
0.054426927119493484,
0.06267908215522766,
0.002990896347910166,
0.023409642279148102,
-0.03978544846177101,
-0.07695367187261581,
-0.0020594217348843813,
0.017903020605444908,
-0.027039803564548492,
-0.024049807339906693,
-0.03899383917450905,
0.028558487072587013,
0.06601744890213013,
-0.01654774136841297,
-0.030954333022236824,
-0.010740919969975948,
-0.018513748422265053,
0.03671855106949806,
-0.05004394054412842,
-0.030804472044110298,
0.04576153680682182,
-0.04694011062383652,
0.012744772247970104,
0.01586749218404293,
0.054852426052093506,
-0.060067176818847656,
0.01700134016573429,
0.09949322789907455,
0.007655499968677759,
0.005974187981337309,
0.003990555182099342,
-0.07557214796543121,
0.06755131483078003,
0.04653492197394371,
0.05726317688822746,
0.010056893341243267,
0.04201271012425423,
-0.003681949805468321,
0.010632113553583622,
0.06366012245416641,
-0.04748019576072693,
0.0020489960443228483,
0.0050310054793953896,
0.08679870516061783,
-0.0041907308623194695,
-0.040951117873191833,
-0.04408605024218559,
-0.017588583752512932,
0.0033443246502429247,
-0.010745236650109291,
-0.011523181572556496,
-0.03046881966292858,
0.04822363704442978,
-0.13298580050468445,
-0.004578390158712864,
0.011113302782177925,
0.0013167719589546323,
-0.04189793020486832,
0.12263999134302139,
0.08505275100469589,
-0.051290519535541534,
0.0542588010430336,
-0.01849261485040188,
-0.014740586280822754,
0.03625579550862312,
-0.001725296606309712,
0.03333880379796028,
-0.06308111548423767,
0.0602731928229332,
-0.026850173249840736,
0.006714197341352701,
0.01218462735414505,
-0.045949146151542664,
0.009516671299934387,
-0.018292006105184555,
-0.02445436455309391,
-0.06930416077375412,
-0.14894740283489227,
0.004329126328229904,
-0.049185048788785934,
0.030404534190893173,
0.036117106676101685,
4.0150151985907435e-33,
0.04478376358747482,
-0.06401816755533218,
-0.03563285991549492,
-0.009473815560340881,
-0.03581715747714043,
-0.05332759767770767,
0.09055564552545547,
-0.021004466339945793,
0.06042443588376045,
0.08217908442020416,
0.051998816430568695,
0.031276922672986984,
-0.10535833239555359,
0.0902499109506607,
0.015874715521931648,
0.014519589953124523,
-0.015476781874895096,
-0.04906008392572403,
-0.060886185616254807,
-0.07303778827190399,
0.002021660329774022,
0.009733001701533794,
-0.018196333199739456,
-0.02482287585735321,
0.026440149173140526,
0.016624603420495987,
0.10110694169998169,
-0.028948895633220673,
-0.018859991803765297,
0.010773643851280212,
-0.013699234463274479,
0.005693783983588219,
0.07068021595478058,
0.0964474081993103,
-0.054688047617673874,
-0.0008208286599256098,
-0.03499361500144005,
-0.0539737343788147,
-0.033342644572257996,
-0.00004128229193156585,
0.04387679696083069,
0.049744728952646255,
0.06870972365140915,
0.0018725832924246788,
0.0069092451594769955,
0.04761943966150284,
-0.011195573955774307,
0.07106082886457443,
-0.014766723848879337,
0.04475519433617592,
-0.041905175894498825,
-0.04578183591365814,
0.11404100060462952,
0.03251826763153076,
0.10876958072185516,
0.005719325039535761,
-0.02013307996094227,
-0.026890845969319344,
0.030582726001739502,
-0.022397849708795547,
-0.006250130478292704,
-0.04537055641412735,
-0.035053011029958725,
0.036550313234329224,
-0.0033962889574468136,
0.10937516391277313,
0.027211138978600502,
-0.07307983189821243,
0.0549747534096241,
-0.0019711267668753862,
-0.025491835549473763,
0.06524965167045593,
-0.05845014005899429,
-0.03622432053089142,
-0.09482953697443008,
-0.05360545590519905,
-0.07465182989835739,
0.07437312602996826,
0.04448596388101578,
-0.08333061635494232,
-0.09138940274715424,
0.0158234890550375,
-0.06618744879961014,
0.023584449663758278,
-0.049135368317365646,
-0.058287039399147034,
-0.006465033628046513,
0.010112729854881763,
0.1266806423664093,
0.03758028522133827,
0.02957303635776043,
0.011465057730674744,
-0.0898774191737175,
0.0426461398601532,
0.028419071808457375,
4.912663019028555e-32,
0.014796658419072628,
-0.0034140339121222496,
-0.030011052265763283,
0.01616032049059868,
-0.08292167633771896,
0.02487213723361492,
0.013692365027964115,
-0.043428875505924225,
-0.01585960015654564,
-0.039769887924194336,
0.08188962936401367,
0.033553943037986755,
-0.07498963922262192,
0.045846715569496155,
-0.09418832510709763,
0.01433255523443222,
-0.011042354628443718,
-0.039188504219055176,
0.0017091479385271668,
-0.06986252963542938,
-0.02224617637693882,
0.0316770076751709,
-0.04826248809695244,
-0.003353044157847762,
-0.02968265861272812,
0.058167148381471634,
0.010465733706951141,
0.011826702393591404,
-0.009417061693966389,
-0.06759680062532425,
0.017758769914507866,
0.000512800004798919,
-0.027312492951750755,
-0.06556867808103561,
0.024769147858023643,
-0.01741604320704937,
-0.03492994233965874,
0.02071661874651909,
-0.026170386001467705,
-0.01098996214568615,
-0.0038887590635567904,
-0.02715875580906868,
-0.0347561277449131,
-0.022608187049627304,
-0.09677693247795105,
0.0218987874686718,
-0.015555786900222301,
-0.06846702098846436,
0.012359973974525928,
0.033118028193712234,
-0.05506208911538124,
0.035590726882219315,
0.11213173717260361,
0.06566839665174484,
0.0233719814568758,
-0.05748538672924042,
-0.023507630452513695,
0.010065601207315922,
0.019093932583928108,
0.007792573422193527,
-0.08174635469913483,
0.10867490619421005,
-0.05003799498081207,
-0.011688514612615108
] |
30
Chapter 2
1 Sketch the gra
phs of the following equations. For each graph, show the coordinates of the point(s)
where the graph crosses the coordinate axes, and write down the coordinate of the turning point
and the equation of the line of symmetry.
a y =
x2 β 6x + 8 b y = x2 + 2x β 15 c y = 25 β x2 d y = x2 + 3x + 2
e y =
βx2 + 6x + 7 f y = 2x2 + 4x + 10 g y = 2x2 + 7x β 15 h y = 6x2 β 19x + 10
i y =
4 β 7x β 2x2 j y = 0.5x2 + 0.2x + 0.02
2 These sketches are gr
aphs of quadratic functions of the form ax2 + bx + c.
Find the values of a, b and c for each function.
a y
x15
5 3y = f(x) b y
x10
β2
5y = g(x)
c y
x
β183 β3
y = h(x) d y
x
β14β1
y = j(x)
3 The graph of y = ax2 + bx + c has a minimum at (5, β3) and passes through (4, 0).
Find the values of a, b and c. (3 marks)P
E/PExercise 2F
2.5 The discriminant
If you square any real number, the result is greater than or equal to 0. This means that if y is negative,
ββ―__
y cannot be a real number. Look at the quadratic formula:
x = βb Β± ββ―_______ b 2 β 4ac ____________ 2a
β For the quadratic function f( x) = ax2 + bx + c, the expression b2 β 4ac is called the
discriminant. The value of the discriminant shows how many roots f( x) has:
β’If b2 β 4ac . 0 then f( x) has two distinct real roots.
β’If b2 β 4ac = 0 then f( x) has one repeated root.
β’If b2 β 4ac , 0 then f( x) has no real roots.Check your answers
by substituting values into the function. In part c the graph passes through (0, β 18), so h(0)
should be β 18.Problem-solving
If the value under the square root sign is negative, x cannot be a real number and there are no real solutions. If the value under the square root is equal to 0, both solutions will be the same.
|
[
-0.06113474816083908,
0.04950324818491936,
-0.027065442875027657,
-0.010381592437624931,
-0.04677857831120491,
-0.02307600900530815,
0.02153606154024601,
0.03756258264183998,
-0.08058635145425797,
0.05030027776956558,
0.03649017587304115,
-0.04662029817700386,
-0.022744959220290184,
-0.013766401447355747,
-0.056469354778528214,
-0.09546568989753723,
-0.05246398225426674,
-0.008096099831163883,
-0.03152958303689957,
0.025079764425754547,
0.007128439843654633,
-0.07646285742521286,
-0.03986573964357376,
-0.012971181422472,
0.03780858591198921,
-0.042440783232450485,
0.040294989943504333,
-0.023987922817468643,
-0.02487611584365368,
-0.01470930501818657,
-0.005766674876213074,
0.006907723378390074,
-0.011600100435316563,
-0.04310416057705879,
0.07254455238580704,
0.0314917117357254,
-0.009469947777688503,
-0.040355514734983444,
0.046693090349435806,
-0.010504278354346752,
0.02395244874060154,
0.06167567893862724,
-0.01455721165984869,
-0.007532677613198757,
0.060386259108781815,
0.012072158046066761,
-0.0367683544754982,
0.045865245163440704,
0.01009660679847002,
-0.01438592653721571,
0.014971219003200531,
-0.00280798994936049,
-0.044600486755371094,
0.04441868141293526,
0.0007434819126501679,
0.02319771610200405,
0.05117852985858917,
0.04816196486353874,
0.021510275080800056,
0.06678353995084763,
0.04178331419825554,
0.0023644650354981422,
-0.016282128170132637,
0.0677969828248024,
-0.010707569308578968,
0.017925303429365158,
0.07751762121915817,
-0.08277460187673569,
-0.017070891335606575,
0.019915718585252762,
-0.12397200614213943,
0.013902834616601467,
-0.02254996821284294,
-0.06894278526306152,
0.07375293225049973,
0.003940991126000881,
0.0551450252532959,
-0.03615957126021385,
-0.010391458868980408,
-0.05632437393069267,
0.023726867511868477,
-0.030983146280050278,
0.10240219533443451,
0.03540245071053505,
0.05447093024849892,
0.008318749256432056,
-0.04786783829331398,
-0.10012978315353394,
0.044677168130874634,
0.005585954524576664,
0.04582744091749191,
0.02997015044093132,
-0.04572677984833717,
-0.05537242814898491,
0.023515425622463226,
-0.052849337458610535,
0.027484511956572533,
-0.04527606442570686,
0.054242685437202454,
0.09576832503080368,
-0.04394232854247093,
0.06784170120954514,
0.04902989789843559,
-0.028966257348656654,
0.03402150049805641,
-0.06604210287332535,
0.05009669065475464,
-0.06683007627725601,
0.04051942750811577,
-0.025133710354566574,
-0.005532579030841589,
-0.08940844982862473,
0.06937139481306076,
0.05174674838781357,
0.03255995362997055,
-0.06658857315778732,
0.06852617859840393,
0.047163672745227814,
-0.025492912158370018,
-0.041035521775484085,
0.06949874758720398,
-0.10677149891853333,
-0.008708261884748936,
-0.062273602932691574,
-0.04062780365347862,
-0.07166092097759247,
-0.078255794942379,
0.024540403857827187,
-0.005785125307738781,
0.027402374893426895,
0.024577954784035683,
-0.06357541680335999,
-0.06665036082267761,
-0.021809479221701622,
-0.06042703241109848,
-0.0902717262506485,
-0.06718065589666367,
0.03322480991482735,
-0.0065680998377501965,
-0.033803679049015045,
-0.0011251881951466203,
0.10902630537748337,
-0.06574273109436035,
0.047662392258644104,
-0.010993911884725094,
0.010614914819598198,
-0.04783521592617035,
-0.01286116149276495,
-0.008750648237764835,
-0.012819760479032993,
0.01903543993830681,
0.012765499763190746,
0.005866377614438534,
0.051080767065286636,
-0.02912568487226963,
0.04109238460659981,
-0.06344740092754364,
-0.031270261853933334,
0.04048382118344307,
0.017560068517923355,
-0.04389570280909538,
0.06790176779031754,
-0.0316796638071537,
0.011196678504347801,
0.0658934935927391,
0.014963777735829353,
0.06813713163137436,
0.05146988481283188,
-0.012123802676796913,
0.08028668910264969,
0.05747314915060997,
-0.03438567370176315,
0.0323016420006752,
0.08129853010177612,
0.058486323803663254,
0.01401938870549202,
0.12358076870441437,
0.037783630192279816,
-0.03363092988729477,
0.019961604848504066,
0.032161857932806015,
-0.06722718477249146,
-0.009730970486998558,
0.02721455506980419,
-0.006550850346684456,
-0.033127687871456146,
-0.07382475584745407,
0.062210939824581146,
-0.0320829413831234,
0.09922393411397934,
-0.0057047889567911625,
-0.018476856872439384,
-0.031987257301807404,
0.0889105498790741,
-0.10668747872114182,
-0.05226413160562515,
0.03410499542951584,
0.03642816096544266,
-0.06523200124502182,
-0.023026039823889732,
0.032427214086055756,
0.04903772845864296,
0.014170724898576736,
0.034990344196558,
0.05501425266265869,
0.027738073840737343,
-0.03851944953203201,
0.036971352994441986,
-0.11244615912437439,
-0.022121770307421684,
0.039073675870895386,
-0.041863854974508286,
-0.08609835058450699,
-0.018468579277396202,
0.08321616053581238,
-0.09529036283493042,
0.01875031180679798,
-0.1308644860982895,
-0.11285002529621124,
-0.03582253307104111,
0.01674714684486389,
0.03616026043891907,
0.04747378081083298,
-2.0759241255781136e-33,
0.02920316904783249,
0.029068754985928535,
-0.05750975385308266,
-0.02543773129582405,
0.055055391043424606,
-0.09196483343839645,
0.06658174097537994,
-0.0006619657506234944,
0.07922407239675522,
0.06591764092445374,
0.09827226400375366,
0.010240624658763409,
-0.04941995441913605,
0.04225579649209976,
0.07167825102806091,
-0.05404597148299217,
0.020673274993896484,
0.07693827152252197,
0.013412993401288986,
-0.029168400913476944,
-0.06116422638297081,
0.019493073225021362,
0.03694065660238266,
0.016260836273431778,
0.010879219509661198,
0.07164163142442703,
0.01186224166303873,
-0.14598335325717926,
-0.06261654198169708,
0.013121573254466057,
-0.010328561067581177,
-0.02911532297730446,
-0.02966541051864624,
0.026697786524891853,
-0.03517633676528931,
-0.08118219673633575,
-0.06295754760503769,
-0.06692762672901154,
-0.08367752283811569,
-0.03931835666298866,
0.02177433855831623,
0.04734421521425247,
0.08404946327209473,
-0.021969199180603027,
-0.043908294290304184,
0.05128313973546028,
0.07465564459562302,
0.042230866849422455,
-0.05234667658805847,
0.014005823992192745,
-0.02591291069984436,
0.0034354364033788443,
0.007212523836642504,
-0.0019018988823518157,
0.050473567098379135,
0.07021638751029968,
0.015930185094475746,
0.04213768243789673,
0.014118161052465439,
-0.08459513634443283,
-0.015135702677071095,
0.0021746873389929533,
0.031023984774947166,
-0.013547218404710293,
-0.06141548976302147,
-0.11867838352918625,
-0.020601222291588783,
-0.07318514585494995,
0.05164254829287529,
0.037669986486434937,
0.01742524653673172,
0.07752522081136703,
-0.007130746729671955,
-0.06225955858826637,
-0.04099544882774353,
-0.06854885071516037,
-0.033908430486917496,
0.0019044751534238458,
0.027423812076449394,
-0.07783293724060059,
-0.09996136277914047,
-0.055768996477127075,
-0.004090868402272463,
-0.01591482199728489,
-0.020258760079741478,
-0.04882718250155449,
0.02772463485598564,
0.043932463973760605,
0.16602301597595215,
-0.0711553618311882,
0.05304085463285446,
-0.017956094816327095,
-0.0022567021660506725,
0.00529573205858469,
0.05778871476650238,
9.400389532212256e-32,
0.003855160204693675,
-0.0017634264659136534,
0.023635463789105415,
0.048461489379405975,
0.05484110116958618,
0.020496416836977005,
0.05578700453042984,
-0.06016271561384201,
-0.08841343224048615,
-0.05458755046129227,
0.060041576623916626,
0.04578695073723793,
-0.0300485547631979,
0.04612080380320549,
-0.061857424676418304,
0.02102270908653736,
-0.05162867531180382,
0.07113496214151382,
-0.0025518445763736963,
0.0016233285423368216,
-0.118577741086483,
-0.031461890786886215,
-0.06264068186283112,
0.052813462913036346,
-0.030045175924897194,
-0.003667656797915697,
0.03236893564462662,
-0.11291951686143875,
-0.022502215579152107,
-0.017421314492821693,
0.05613631382584572,
-0.08461081981658936,
0.02627399004995823,
-0.030324142426252365,
0.004190067294985056,
-0.03638618811964989,
-0.07343287020921707,
0.05203777179121971,
0.004953319672495127,
-0.009807245805859566,
-0.02723652683198452,
0.04815074801445007,
-0.021897917613387108,
0.004079813603311777,
-0.06095001474022865,
0.07858101278543472,
0.015055370517075062,
-0.04103982448577881,
0.027797289192676544,
0.019990265369415283,
-0.07817584276199341,
0.026993650943040848,
0.025555472820997238,
-0.09174661338329315,
-0.03386016935110092,
-0.009762756526470184,
-0.02085152268409729,
0.012644749134778976,
-0.04037855565547943,
0.006583987735211849,
0.01613841950893402,
0.1203281432390213,
-0.04071725532412529,
0.031998809427022934
] |
31Quadratics
You can use the discriminant to check the shape of sketch graphs.
Below are some graphs of y = f(x) where f(x) = ax2 + bx + c.
a . 0
y
x O
y
x O y
x O
b2 β 4ac . 0 b2 β 4ac = 0 b2 β 4ac , 0
Two distinct real roots One repeated r oot No real roots
a
, 0
y
x O
y
x O
y
x O
Find the range of values of k for which x2 + 4x + k = 0 has two distinct real solutions.Example 14
x2 + 4x + k = 0
Here a = 1, b = 4 and c = k .
For two real solutions, b2 β 4 ac . 0
42 β 4 Γ 1 Γ k . 0
16 β 4 k . 0
16 . 4 k
4 . k
So k , 4This statement involves an inequality, so your
answer will also be an inequality.Find the values of k for which f(x) = x2 + kx + 9 has equal roots.Example 13
x2 + kx + 9 = 0
Here a = 1, b = k and c = 9
For equal roots, b2 β 4 ac = 0
k2 β 4 Γ 1 Γ 9 = 0
k2 β 36 = 0
k2 = 36
so k = Β± 6Use the condition given in the question to write a
statement about the discriminant.Problem-solving
Substitute for a, b and c to get an equation with one unknown.
Solve to find the values of k.
For any value of k less than 4, the equation will have 2 distinct real solutions.
Explore how the value of the
di
scriminant changes with k using GeoGebra.Online
|
[
0.0005855903145857155,
0.00960448570549488,
0.002563233021646738,
-0.002069014823064208,
-0.04189464822411537,
0.019326893612742424,
-0.120516836643219,
0.06300268322229385,
-0.11026746034622192,
0.007404150906950235,
-0.03229108825325966,
-0.08342619240283966,
-0.05215401574969292,
0.0336676724255085,
-0.0168608445674181,
0.009300574660301208,
-0.07191432267427444,
0.05392743647098541,
0.0007809425587765872,
-0.0038931500166654587,
0.009010101668536663,
-0.09058167040348053,
-0.06487173587083817,
-0.06461580097675323,
0.08138423413038254,
-0.05851614847779274,
0.040025390684604645,
0.007916604168713093,
-0.0001964126422535628,
-0.01476915180683136,
-0.0013599260710179806,
0.0698072612285614,
0.01401728205382824,
-0.0615486204624176,
0.07213031500577927,
-0.05687858536839485,
0.02245721034705639,
0.07298033684492111,
0.0714183896780014,
-0.013966686092317104,
-0.04182590916752815,
0.05261661484837532,
0.02775638923048973,
0.020718030631542206,
0.023764345794916153,
-0.04113299027085304,
-0.03501923382282257,
0.004529450088739395,
0.1129024401307106,
0.01084593404084444,
0.05004067346453667,
-0.004673573188483715,
-0.06298904865980148,
0.06357993930578232,
0.019681880250573158,
0.04749336466193199,
-0.0598280243575573,
-0.08515022695064545,
0.024773377925157547,
0.03732632100582123,
-0.015068665146827698,
0.04785313829779625,
0.04197760671377182,
0.04152445122599602,
0.04947149381041527,
0.022005606442689896,
0.10348788648843765,
-0.050680581480264664,
0.023657500743865967,
0.027855822816491127,
-0.003990106750279665,
0.02736942656338215,
-0.025167204439640045,
-0.029774151742458344,
0.0009425317985005677,
0.023085176944732666,
-0.04267289862036705,
0.08032723516225815,
-0.0058607966639101505,
-0.04031897336244583,
-0.10124063491821289,
-0.009237085469067097,
0.11989782750606537,
-0.004639614373445511,
-0.029741128906607628,
-0.03703559190034866,
0.05931469798088074,
0.07673672586679459,
-0.004034033976495266,
0.002719805808737874,
0.003538690973073244,
0.08928175270557404,
-0.05408254265785217,
-0.08686513453722,
0.015071950852870941,
-0.06474889069795609,
0.08068470656871796,
-0.05014082416892052,
0.07290159910917282,
0.04693252220749855,
-0.01904335245490074,
-0.0274764783680439,
0.03796059638261795,
0.025609763339161873,
0.03864095360040665,
0.018525632098317146,
0.02159663662314415,
-0.04314981773495674,
-0.03145073726773262,
-0.06067943572998047,
-0.07250630110502243,
-0.04715995490550995,
0.013287577778100967,
0.04056752473115921,
0.0905333012342453,
-0.03392033278942108,
0.06242351979017258,
-0.01765601523220539,
0.0261493269354105,
-0.059127502143383026,
0.04520422965288162,
0.0014736310113221407,
-0.007487480528652668,
-0.07126931101083755,
-0.0054252720437943935,
-0.03906014561653137,
-0.06763455271720886,
0.004161940887570381,
-0.009203653782606125,
-0.055138103663921356,
0.0549849197268486,
0.0296404417604208,
-0.006239984184503555,
-0.049149297177791595,
-0.04216181859374046,
-0.02903696335852146,
-0.00039470879710279405,
0.13012702763080597,
-0.06452503800392151,
-0.019593385979533195,
-0.05279230698943138,
0.054587796330451965,
0.0599253810942173,
0.03598229959607124,
0.025871792808175087,
-0.08910661935806274,
-0.07679827511310577,
-0.019667532294988632,
-0.009258871898055077,
-0.060315877199172974,
0.018116647377610207,
0.04602837190032005,
0.06237559765577316,
0.16872037947177887,
-0.00335642765276134,
-0.03512179106473923,
-0.0035946157295256853,
0.01105172373354435,
0.004292793571949005,
0.042201898992061615,
0.043299395591020584,
0.0030848579481244087,
-0.0244684349745512,
0.006105088163167238,
-0.0293632410466671,
0.015739139169454575,
-0.016803260892629623,
-0.011129132471978664,
0.008004799485206604,
-0.08262085169553757,
0.035195522010326385,
0.03045077808201313,
0.055273935198783875,
0.09161730110645294,
-0.019429393112659454,
0.006905419752001762,
-0.016129145398736,
0.07403059303760529,
0.00787313748151064,
0.010434526950120926,
0.000763667980208993,
-0.006350378040224314,
-0.047326743602752686,
-0.028306176885962486,
0.01121952198445797,
-0.034969158470630646,
-0.013057962991297245,
-0.04722726717591286,
-0.03989269956946373,
-0.026390302926301956,
-0.026681499555706978,
-0.029936032369732857,
-0.14547288417816162,
0.07578270882368088,
-0.14401325583457947,
-0.020603878423571587,
-0.008489946834743023,
-0.02615417167544365,
-0.053968001157045364,
0.055037692189216614,
0.021689213812351227,
0.0021350004244595766,
0.011135531589388847,
-0.004534492269158363,
-0.023200644180178642,
0.062136225402355194,
-0.05442620441317558,
0.048123035579919815,
-0.05755826458334923,
0.002731597749516368,
-0.09055714309215546,
-0.031052304431796074,
-0.016815511509776115,
-0.08801497519016266,
-0.010638711974024773,
-0.09159142524003983,
0.049862124025821686,
-0.006492041517049074,
-0.11980011314153671,
0.006484700366854668,
-0.03808610513806343,
-0.07311085611581802,
0.01576807163655758,
-9.198319702317203e-33,
0.0022801628801971674,
-0.03554874286055565,
-0.02252276800572872,
0.035923343151807785,
-0.06814191490411758,
-0.005313171073794365,
0.057870715856552124,
-0.05261184647679329,
0.09897961467504501,
-0.022498084232211113,
0.11499369144439697,
0.04895439371466637,
-0.054068803787231445,
0.024377595633268356,
-0.021246137097477913,
0.008009890094399452,
-0.05609281361103058,
0.012310013175010681,
0.033991970121860504,
-0.046348899602890015,
0.06690730899572372,
0.05675438418984413,
-0.02488791011273861,
-0.04221101477742195,
0.019654622301459312,
-0.0012012931983917952,
0.05927165597677231,
-0.025074873119592667,
-0.06462819129228592,
0.040944166481494904,
-0.06137947738170624,
-0.010456429794430733,
-0.006285142153501511,
0.013342366553843021,
-0.021925602108240128,
-0.085410475730896,
0.02074335142970085,
-0.02868846245110035,
0.055178720504045486,
-0.004890533164143562,
-0.05148383975028992,
0.053318846970796585,
0.042364753782749176,
0.03913536295294762,
0.036132968962192535,
0.08411695063114166,
0.07515296339988708,
0.016280071809887886,
-0.03428121656179428,
0.015497074462473392,
0.012696377001702785,
-0.029369600117206573,
0.11263972520828247,
0.02392178773880005,
0.10473272949457169,
-0.025592230260372162,
-0.028102189302444458,
0.0015913813840597868,
-0.0003233138704672456,
-0.006979334633797407,
-0.04950522258877754,
-0.10831111669540405,
-0.004914844408631325,
0.06954929977655411,
-0.10401492565870285,
0.016117384657263756,
-0.0330590084195137,
-0.073318712413311,
0.015197623521089554,
0.010476348921656609,
0.025818957015872,
0.02230602130293846,
-0.0644053965806961,
-0.0028494619764387608,
-0.1147749200463295,
-0.01477226335555315,
-0.021042142063379288,
0.05190865322947502,
0.028180673718452454,
-0.042655378580093384,
-0.07480677217245102,
0.04676565155386925,
0.012940509244799614,
0.03527233749628067,
-0.05176810175180435,
0.053793348371982574,
-0.012790692038834095,
0.0782654732465744,
0.01978887803852558,
0.024962028488516808,
0.06305286288261414,
0.039969515055418015,
-0.12593482434749603,
0.04660570248961449,
0.06898302584886551,
1.034313482607738e-31,
-0.034984201192855835,
0.08156536519527435,
-0.0008023866103030741,
-0.019475972279906273,
0.005541478283703327,
0.011463621631264687,
0.0319642536342144,
-0.036463700234889984,
-0.026983866468071938,
-0.03976600244641304,
0.051832590252161026,
0.007991787977516651,
-0.1041397899389267,
0.01484126877039671,
-0.01750214397907257,
0.057192858308553696,
-0.01263553649187088,
0.0483412928879261,
-0.05922191962599754,
-0.05346432700753212,
-0.0969465896487236,
-0.002981304656714201,
-0.023979512974619865,
0.07549910247325897,
-0.0484754778444767,
0.05308305099606514,
-0.07499511539936066,
-0.0719175860285759,
-0.04364342242479324,
0.011058702133595943,
-0.009560290724039078,
0.014538903720676899,
0.033421777188777924,
-0.0400598905980587,
0.07668017596006393,
-0.017085302621126175,
-0.09650050103664398,
0.02379714883863926,
-0.05135180801153183,
-0.04547431319952011,
0.006486350204795599,
-0.06553234159946442,
-0.04328538849949837,
-0.0463755838572979,
-0.0886232927441597,
-0.04430333524942398,
0.0568377822637558,
-0.06987938284873962,
0.07542571425437927,
0.060336414724588394,
-0.06575283408164978,
0.05937676876783371,
-0.009802998043596745,
0.056531574577093124,
0.03708377107977867,
-0.0277915857732296,
0.004507995676249266,
-0.06919471174478531,
-0.018717629835009575,
-0.02397821471095085,
-0.017466848716139793,
0.10498074442148209,
-0.05910303443670273,
-0.011143899522721767
] |
32
Chapter 2
1 a Calcula
te the value of the discriminant for each of these five functions:
i f(x)
= x2 + 8x + 3 ii g(x) = 2x2 β 3x + 4 iii h(x) = βx2 + 7x β 3
iv j(x)
= x2 β 8x + 16 v k(x ) = 2x β 3x2 β 4
b Using your answ
ers to part a, match the same five functions to these sketch graphs.
i
x Oy ii
x Oy iii
Oy iv
x Oy v
x Oy
2 Find the values of k for which x2 + 6x + k = 0 has two real solutions. (2 marks)
3 Find the value of
t for which 2x2 β 3x + t = 0 has exactly one solution. (2 marks)
4 Given tha
t the function f(x ) = sx2 + 8x + s has equal roots, find the value of the positive
constant s. (2 marks)
5 Find the range of v
alues of k for which 3x2 β 4x + k = 0 has no real solutions. (2 marks)
6 The function g(x)
= x2 + 3px + (14p β 3), where p is an integer, has two equal roots.
a Find the value of
p. (2 marks)
b For this va
lue of p, solve the equation x2 + 3px + (14p β 3) = 0. (2 marks)
7 h(x)
= 2x2 + (k + 4)x + k, where k is a real constant.
a Find the discriminant of h(x ) in ter
ms of k . (3 marks)
b Hence or otherwise, pro
ve that h(x ) has two distinct
real roots for all values of k . (3 marks)E/P
E/P
E/P
E/P
E/P
E/PExercise 2G
2.6 Modelling with quadratics
A mathematical model is a mathematical description of a real-life situation. Mathematical
models use the language and tools of mathematics to represent and explore real-life patterns and relationships, and to predict what is going to happen next.
Models can be simple or complicated, and their results can be approximate or exact. Sometimes a model
is only valid under certain circumstances, or for a limited range of inputs. You will learn more about how models involve simplifications and assumptions in Statistics and Mechanics.
Quadratic functions can be used to model and explore a range of practical contexts, including
projectile motion.a Prove that, if the values of a and c are given and non-zero, it is always possible to choose a value of
b so that f(x) = ax2 + bx + c has distinct real roots.
b Is it alway
s possible to choose a value of b so that f(x) has equal roots? Explain your answer.ChallengeIf a question part says βhence or
otherwiseβ it is usually easier to use your answer to the previous question part.Problem-solving
|
[
0.0022119900677353144,
0.06606264412403107,
-0.009269393049180508,
-0.10749517381191254,
0.015220246277749538,
0.07941683381795883,
-0.006387798581272364,
0.05260474607348442,
-0.09636147320270538,
0.025897622108459473,
0.05179930478334427,
-0.06933324784040451,
-0.059752628207206726,
0.02417854592204094,
-0.05248495936393738,
-0.016936490312218666,
-0.012824212200939655,
0.05259928107261658,
-0.09028606116771698,
-0.05522711202502251,
0.02347017638385296,
0.027451198548078537,
-0.08773759752511978,
-0.103598453104496,
0.0819358304142952,
-0.06044714152812958,
0.060527503490448,
0.025561412796378136,
-0.009214173071086407,
-0.09022253751754761,
-0.048240866512060165,
0.00230479147285223,
0.022548042237758636,
-0.02726725861430168,
0.01845630072057247,
-0.0037512045819312334,
0.0025030956603586674,
0.06094622239470482,
0.09332163631916046,
-0.04309453070163727,
-0.005783773958683014,
0.011404801160097122,
-0.0041715470142662525,
0.06023510545492172,
0.02770310826599598,
-0.05601557344198227,
-0.035270676016807556,
0.026542119681835175,
0.0025520126800984144,
-0.010256475768983364,
-0.0062482780776917934,
0.007611454464495182,
-0.03745526075363159,
0.04909342899918556,
0.04297387972474098,
-0.046514105051755905,
0.02013837732374668,
0.013989988714456558,
0.019028866663575172,
0.05165288224816322,
-0.026839042082428932,
0.02177196368575096,
-0.009915526956319809,
0.06562826782464981,
0.0038685789331793785,
0.05753330513834953,
0.10101503878831863,
-0.04710175096988678,
-0.025151820853352547,
0.04251718893647194,
-0.07637546211481094,
-0.006685805972665548,
-0.08137752115726471,
-0.06847452372312546,
0.006731588393449783,
0.035708237439394,
0.015530215576291084,
-0.011463018134236336,
-0.029786977916955948,
-0.0253459420055151,
-0.022088391706347466,
0.0278167724609375,
0.032037507742643356,
0.07586032897233963,
0.0017353853909298778,
-0.015816202387213707,
-0.023904601112008095,
0.041064802557229996,
0.01869218610227108,
-0.017380105331540108,
-0.03294192999601364,
0.02803475596010685,
-0.014307432807981968,
-0.056497666984796524,
-0.012370476499199867,
-0.05334791913628578,
0.026611408218741417,
-0.0714307427406311,
0.09781099110841751,
0.09430573135614395,
-0.018298231065273285,
-0.0381552055478096,
0.06986097991466522,
0.062366120517253876,
-0.03249315172433853,
-0.04809537157416344,
0.03219064697623253,
-0.05662096291780472,
-0.04641924798488617,
-0.11952602118253708,
-0.04073786735534668,
-0.08092872053384781,
-0.002370106987655163,
0.04719963297247887,
0.11157792806625366,
-0.007994160987436771,
0.12715335190296173,
-0.08321625739336014,
0.012196301482617855,
-0.06026254966855049,
0.009614101611077785,
-0.0072827422991395,
0.028501277789473534,
-0.05120796337723732,
-0.0010716236429288983,
-0.037604279816150665,
-0.027557725086808205,
-0.027233600616455078,
0.0034649288281798363,
-0.029302233830094337,
0.031128911301493645,
-0.00019022983906324953,
-0.008172105997800827,
-0.051581285893917084,
-0.10058484226465225,
-0.04689178988337517,
-0.07972530275583267,
0.11258683353662491,
-0.053722795099020004,
0.03522154316306114,
-0.07193303853273392,
0.10969352722167969,
0.016771864145994186,
0.08135028183460236,
0.05076643079519272,
-0.06961938738822937,
-0.10351329296827316,
-0.03462393954396248,
0.0758255124092102,
0.028609471395611763,
-0.025390496477484703,
0.019056208431720734,
0.05086443945765495,
0.1460876315832138,
0.02406259998679161,
-0.08283264189958572,
0.012328407727181911,
-0.03923505172133446,
0.0030150667298585176,
-0.038351621478796005,
0.01998639665544033,
0.011556779034435749,
0.05060041323304176,
0.09209245443344116,
0.06919174641370773,
0.02321055717766285,
0.05005236715078354,
0.009527680464088917,
-0.017396744340658188,
-0.06923808157444,
0.007622553035616875,
-0.017615608870983124,
0.0071494909934699535,
0.027345305308699608,
0.031572259962558746,
-0.018529901280999184,
0.08645743131637573,
0.07215627282857895,
0.009944590739905834,
0.022358667105436325,
0.053443145006895065,
-0.05702263116836548,
-0.04355233535170555,
0.018514905124902725,
-0.019566256552934647,
-0.0035429794806987047,
-0.02342209778726101,
0.0160236619412899,
-0.04567163810133934,
-0.027290811762213707,
-0.013059833087027073,
-0.07280067354440689,
-0.05780281871557236,
0.06381251662969589,
-0.10789815336465836,
-0.05318464711308479,
-0.020511655136942863,
-0.05361412838101387,
-0.11228450387716293,
0.06692589074373245,
0.09392834454774857,
-0.03797699511051178,
0.1146421730518341,
0.01592167466878891,
-0.04187377914786339,
-0.003938011359423399,
-0.1451231837272644,
-0.009152228944003582,
-0.05507998540997505,
0.04329027235507965,
-0.09933912754058838,
0.024324946105480194,
-0.08834654092788696,
-0.04392533749341965,
0.04961002990603447,
-0.061145734041929245,
0.06224944442510605,
-0.04408813640475273,
-0.08936057984828949,
-0.0191950760781765,
-0.01205072458833456,
-0.07777143269777298,
0.07251689583063126,
-1.5018815245244284e-32,
-0.032374318689107895,
0.001118660788051784,
-0.010875978507101536,
-0.02812156453728676,
-0.026021480560302734,
-0.05856514722108841,
0.06768352538347244,
-0.09172894060611725,
0.033476658165454865,
0.020236801356077194,
0.03983504697680473,
0.04342782124876976,
-0.06966805458068848,
-0.01653123088181019,
-0.07007744908332825,
0.013438739813864231,
-0.05363733693957329,
0.039314672350883484,
0.008858892135322094,
-0.015018409118056297,
0.03220430389046669,
0.05676715448498726,
0.05505193769931793,
-0.008481121622025967,
-0.01180652529001236,
-0.01528196781873703,
0.03474430367350578,
-0.09316013008356094,
-0.010244970209896564,
0.031168073415756226,
-0.007006514351814985,
-0.01613800786435604,
0.038381919264793396,
0.021760644391179085,
0.007910300977528095,
-0.06906354427337646,
-0.01890481822192669,
-0.04435232654213905,
0.024034690111875534,
-0.0066189635545015335,
0.04961482435464859,
0.02993830293416977,
0.02771412767469883,
0.022746579721570015,
0.026627423241734505,
0.06989046931266785,
0.046277500689029694,
-0.021589413285255432,
0.015484443865716457,
0.014566418714821339,
-0.008212226442992687,
-0.06816648691892624,
0.034312497824430466,
0.01612364687025547,
0.058630697429180145,
-0.046234969049692154,
0.06091291457414627,
-0.05438840761780739,
0.061278022825717926,
0.007873292081058025,
-0.05096428468823433,
-0.0489676333963871,
-0.08337155729532242,
0.06622451543807983,
-0.06984342634677887,
-0.10148607194423676,
-0.06300202012062073,
-0.012957784347236156,
0.04544777050614357,
-0.019118426367640495,
0.05877690017223358,
-0.002008444629609585,
0.018651966005563736,
-0.05174433812499046,
-0.0676596462726593,
-0.07258811593055725,
0.0023342689964920282,
0.009611339308321476,
-0.008222324773669243,
-0.019298970699310303,
-0.050064440816640854,
0.011621115729212761,
0.03171277791261673,
-0.0006232271553017199,
-0.020726345479488373,
0.04089002311229706,
0.07091684639453888,
0.09242074191570282,
0.0421452596783638,
-0.04308216646313667,
0.0414559468626976,
0.05524776503443718,
-0.03573588281869888,
-0.053952064365148544,
0.05024069547653198,
1.2234681560703064e-31,
-0.07011586427688599,
0.03242427855730057,
-0.007550401613116264,
0.053552523255348206,
-0.03445884585380554,
-0.08849432319402695,
-0.027466177940368652,
-0.062376972287893295,
-0.0337924063205719,
-0.004969105124473572,
0.1526649296283722,
0.017261913046240807,
-0.05020183324813843,
-0.0065839956514537334,
0.014110973104834557,
0.06107715889811516,
-0.007321506273001432,
0.03918277472257614,
-0.0204253401607275,
-0.043087251484394073,
-0.10820453613996506,
0.007706061005592346,
-0.02958489954471588,
0.03005659580230713,
-0.03039540722966194,
0.04210669547319412,
-0.0590520016849041,
-0.09137682616710663,
0.024771446362137794,
0.009185983799397945,
0.01741994544863701,
0.06583265215158463,
0.03705810755491257,
-0.060385432094335556,
0.035525355488061905,
0.019801534712314606,
-0.06754831969738007,
0.008715893141925335,
-0.0060613201931118965,
0.0018612866988405585,
-0.05954489856958389,
-0.032171133905649185,
-0.004332712385803461,
0.00899647455662489,
-0.038926005363464355,
-0.05882301926612854,
0.0695502758026123,
-0.04370330274105072,
0.04459583759307861,
0.0360523946583271,
-0.035067878663539886,
0.0032259339932352304,
-0.04713228717446327,
-0.028988877311348915,
0.05660488083958626,
-0.023063281551003456,
0.007704072631895542,
-0.07200922816991806,
-0.036616794764995575,
-0.06647131592035294,
-0.0033609485253691673,
0.07852516323328018,
-0.07204211503267288,
0.0024518026039004326
] |
33Quadratics
A spear is thrown over level ground from the top of a tower.
The height, in metres, of the spear above the ground after t seconds is modelled by the function:
h(t) = 12.25 + 14.7t β 4.9t2, t > 0
a Interpret the meaning of the constant ter
m 12.25 in the model.
b After how many seconds does the spear hit the gr
ound?
c Write h(t
) in the form A β B(t β C)2, where A, B and C are constants to be found.
d Using your answ
er to part c or otherwise, find the maximum height of the spear above the
ground, and the time at which this maximum height is reached.Example 15
a The tower is 12.25 m tal l, since
this is the height at time 0.
b Whe
n the spear hits the ground,
the height is equal to 0.
12.25 + 14.7 t β 4.9 t2 = 0
Using the formula, where a = β 4.9,
b = 14.7 and c = 12.25,
t = β14.7 Β± β ___________________ 14. 7 2 β 4 (β4.9 ) (12.25 ) _______________________________ (2 Γ β4.9)
t = β14.7 Β± β ______ 456.19 _________________ β9.8
t = Β β0.679 or t = 3.68 (to 3 s
.f.)
As t > 0, t = 3.68 seconds (to 3 s.f.).
c 12.
25 + 14.7 t β 4.9 t2
= β4.9(t2 β 3 t) + 12.25
= β4.9(( t β 1.5)2 β 2.25) + 12.25
= β4.9(( t β 1.5)2 + 11.025 + 12.25)
= 23.275 β 4.9( t β 1.5)2
So A = 23.275, B = 4.9 and C = 1.5.
d The m
aximum height of the spear is
23.275 metres, 1.5 seconds after
the spear is thrown.Give any non-exact numerical answers correct to
3 significant figures unless specified otherwise.
Always interpret your answers in the context of the model. t is the time after the spear was thrown so it must be positive.Read the question carefully to work out the meaning of the constant term in the context of the model. Here, t = 0 is the time the spear is thrown.Problem-solving
To solve a quadratic, factorise, use the quadratic formula, or complete the square.
4.9(t β 1.5)2 must be positive or 0, so h(t ) < 23.275
for all possible values of t .
The turning point of the graph of this function would be at (1.5, 23.275). You may find it helpful to draw a sketch of the function when working through modelling questions.
Explore the trajectory of the
sp
ear using GeoGebra.Online
|
[
0.08113740384578705,
0.13460774719715118,
0.029531868174672127,
-0.028292525559663773,
-0.056732237339019775,
-0.13310174643993378,
-0.0005833124159835279,
0.10582779347896576,
-0.03586481511592865,
0.010685348883271217,
0.10464925318956375,
-0.029199572280049324,
-0.04870618134737015,
-0.025464966893196106,
-0.032269615679979324,
0.012912515550851822,
-0.044976867735385895,
-0.01434270292520523,
-0.07938745617866516,
-0.03223753347992897,
-0.0670800730586052,
-0.017370473593473434,
-0.021370967850089073,
0.05786535516381264,
0.05307100713253021,
-0.014557857997715473,
-0.004830623045563698,
0.00197434495203197,
-0.010452468879520893,
-0.01338809635490179,
-0.05567442625761032,
-0.053333982825279236,
0.0064313821494579315,
0.036186836659908295,
-0.007682625204324722,
0.09228190779685974,
0.019082695245742798,
0.05116986855864525,
-0.013184458017349243,
0.02879471331834793,
-0.06979912519454956,
0.0321061946451664,
-0.062059544026851654,
0.007364352699369192,
0.05051153153181076,
-0.0031765284948050976,
-0.07709135115146637,
-0.00692789489403367,
-0.04378887638449669,
0.0025620991364121437,
0.035736750811338425,
0.019101466983556747,
-0.09976893663406372,
0.026759032160043716,
-0.02119631879031658,
-0.025799138471484184,
0.056744325906038284,
0.06368844211101532,
0.0705719143152237,
-0.025937212631106377,
-0.04878581315279007,
0.06640515476465225,
-0.007818968035280704,
0.06489422917366028,
0.03833118826150894,
0.03152081370353699,
-0.03999227657914162,
-0.036903951317071915,
0.01722641848027706,
0.08461020886898041,
-0.07075238227844238,
0.06818849593400955,
-0.042266104370355606,
-0.09283801168203354,
0.02767687290906906,
-0.1270149201154709,
0.026426034048199654,
-0.0032568846363574266,
-0.025551632046699524,
0.03499709069728851,
0.0387350432574749,
-0.03158845007419586,
-0.06062253937125206,
0.03729843720793724,
-0.04022080451250076,
0.04291556403040886,
0.005441131070256233,
0.06530524045228958,
-0.013647234067320824,
0.04920833930373192,
0.03366531431674957,
-0.023432251065969467,
-0.012304609641432762,
0.015440477058291435,
-0.04400352016091347,
0.0047035482712090015,
0.011775481514632702,
-0.0913778766989708,
0.003947894088923931,
0.14976511895656586,
-0.02135043777525425,
-0.008036335930228233,
-0.03373364731669426,
0.06082536280155182,
0.07344989478588104,
-0.004817793611437082,
-0.05046355724334717,
0.037334755063056946,
-0.058498285710811615,
-0.026202546432614326,
-0.002316221594810486,
0.007766271475702524,
0.013052438385784626,
-0.02667098306119442,
0.05995362997055054,
-0.05155373737215996,
-0.003829633817076683,
-0.1156223937869072,
-0.06148248538374901,
-0.0871933177113533,
0.05779307708144188,
-0.009615445509552956,
0.030902395024895668,
-0.05266362801194191,
-0.006986954715102911,
0.03571831062436104,
0.033119529485702515,
0.06575722247362137,
-0.0778191015124321,
0.028423478826880455,
-0.02408306859433651,
-0.13413645327091217,
-0.0277500431984663,
-0.0463009774684906,
-0.04205205291509628,
0.013016964308917522,
-0.00010915124585153535,
0.11097311973571777,
-0.00949896965175867,
-0.055218033492565155,
-0.09014633297920227,
0.03469885140657425,
-0.022122424095869064,
-0.06418805569410324,
0.09124337136745453,
0.0404181033372879,
-0.045284759253263474,
-0.0396253727376461,
-0.04095236957073212,
0.041463837027549744,
-0.011948484927415848,
0.044255275279283524,
0.07880950719118118,
0.0736905112862587,
-0.024933570995926857,
-0.01899273507297039,
-0.030522499233484268,
-0.059473615139722824,
-0.06356441974639893,
-0.05704442784190178,
-0.11167225986719131,
-0.009213538840413094,
0.050795912742614746,
-0.039733726531267166,
0.07265984266996384,
0.007305417209863663,
0.054205521941185,
0.03214537352323532,
-0.0038971961475908756,
-0.0004715556278824806,
0.006270217709243298,
-0.02999299019575119,
-0.08046634495258331,
0.05219326540827751,
0.025413669645786285,
0.040811311453580856,
0.02298811823129654,
0.04421406611800194,
0.043991439044475555,
-0.004729293752461672,
0.03500973805785179,
-0.03577560558915138,
0.061673786491155624,
0.05547633394598961,
-0.002404880942776799,
0.0034380373544991016,
-0.015176624990999699,
0.0328071229159832,
0.015140925534069538,
0.012397278100252151,
0.054785553365945816,
-0.027661887928843498,
-0.0545487105846405,
0.0014506169827654958,
-0.05999831110239029,
-0.01159914955496788,
-0.017322883009910583,
0.05044235289096832,
-0.028898146003484726,
0.02631971798837185,
0.04149291664361954,
0.007454757113009691,
-0.03625841811299324,
0.03222418203949928,
0.03178489953279495,
0.01050015538930893,
-0.003929237835109234,
-0.0631546750664711,
-0.07087630033493042,
0.03991762176156044,
-0.035157520323991776,
-0.020442906767129898,
-0.0509306900203228,
-0.032923758029937744,
0.03199172392487526,
-0.05782869830727577,
-0.05860642343759537,
-0.06783246248960495,
-0.03621687367558479,
0.09890613704919815,
-0.055539775639772415,
-0.039394862949848175,
0.053759340196847916,
8.98014501206871e-33,
0.007349853403866291,
0.09722572565078735,
-0.07062245905399323,
-0.06851982325315475,
0.029513172805309296,
-0.07040566205978394,
-0.01309115905314684,
-0.05141054838895798,
0.036516111344099045,
0.01906745322048664,
-0.0010758517310023308,
0.029371783137321472,
-0.062033239752054214,
-0.03260419890284538,
0.05116359889507294,
-0.11641991883516312,
0.033631760627031326,
-0.004343732260167599,
-0.02330656535923481,
0.02702031284570694,
-0.024255402386188507,
-0.020141979679465294,
0.00798838958144188,
-0.013206243515014648,
-0.013312838040292263,
0.019780214875936508,
-0.048668768256902695,
-0.06707344204187393,
-0.024945681914687157,
-0.024662697687745094,
-0.05507855489850044,
-0.028620118275284767,
0.05526592954993248,
0.0533999539911747,
-0.038690660148859024,
-0.060948483645915985,
0.0559043362736702,
-0.03846392408013344,
0.0048220036551356316,
0.0030017406679689884,
0.11168932914733887,
0.03849833831191063,
0.003506766865029931,
0.023294635117053986,
-0.0382063128054142,
-0.002515921602025628,
-0.042644139379262924,
0.01680293120443821,
0.01414474193006754,
0.01440460979938507,
0.01069556549191475,
-0.05473487824201584,
0.0047555421479046345,
-0.008568990975618362,
0.11853155493736267,
0.04044663906097412,
-0.002991017187014222,
-0.1049823984503746,
-0.004319746978580952,
-0.01350640133023262,
0.029611971229314804,
-0.09405534714460373,
0.020040113478899002,
0.07817757874727249,
-0.05535966902971268,
0.020173119381070137,
-0.009292841888964176,
-0.023834815248847008,
-0.08462037146091461,
0.0890512615442276,
-0.06576220691204071,
0.14457620680332184,
-0.001913200132548809,
-0.07893386483192444,
-0.03754028305411339,
-0.08794102817773819,
-0.021804949268698692,
-0.0177988950163126,
0.047209154814481735,
0.007153165061026812,
-0.020841648802161217,
0.04417033493518829,
0.03929198905825615,
-0.028948429971933365,
-0.10429098457098007,
-0.02028386853635311,
0.06167086958885193,
-0.026937682181596756,
0.06121402978897095,
0.11728078126907349,
0.10580584406852722,
0.04203960672020912,
0.016598770394921303,
-0.02563856728374958,
-0.023064615204930305,
7.491407865375176e-32,
0.007577018812298775,
0.05025428161025047,
-0.04021125286817551,
0.006157684605568647,
0.018195411190390587,
0.12394031137228012,
0.021110406145453453,
-0.05498296767473221,
-0.03565531596541405,
-0.025329895317554474,
0.050627805292606354,
0.04186501353979111,
0.017195014283061028,
0.020444583147764206,
-0.07177358865737915,
-0.013174724765121937,
-0.06391724199056625,
-0.004143138881772757,
-0.04649269953370094,
-0.005451295990496874,
-0.07826496660709381,
0.07305601239204407,
-0.03555229678750038,
0.12986169755458832,
0.008024993352591991,
0.060195621103048325,
-0.05381942540407181,
0.0433298796415329,
-0.024726267904043198,
-0.051448047161102295,
0.09723508358001709,
0.018067054450511932,
0.02578551322221756,
-0.03035554848611355,
-0.009694771841168404,
0.03799276053905487,
0.035987019538879395,
0.0297582745552063,
-0.026772834360599518,
0.08509038388729095,
-0.04178300127387047,
-0.038716476410627365,
-0.07202445715665817,
0.07284603267908096,
-0.045899808406829834,
0.03565150499343872,
0.0019855850841850042,
-0.047991793602705,
0.07135571539402008,
0.0768040344119072,
-0.05483494699001312,
0.047292403876781464,
0.013025971129536629,
-0.03422122821211815,
0.07762914150953293,
0.06861644238233566,
0.018855826929211617,
-0.010389905422925949,
-0.04702245816588402,
-0.0759224146604538,
-0.004731484688818455,
0.02530505508184433,
-0.04392585903406143,
0.04455697536468506
] |
34
Chapter 2
1 The diagram sho
ws a section of a suspension bridge carrying a road over water.
The height of the cables above water level in metres can be modelled by the function
h(x)Β =Β 0.000 12x2 + 200, where x is the displacement in metres from the centre of the bridge.
a Interpret the meaning of the constant ter
m 200 in the model. (1 mark)
b Use the model to find the two v
alues of x at which the height is 346 m. (3 marks)
c Given tha
t the towers at each end are 346 m tall, use your ans
wer to part b to calculate the
length of the bridge to the nearest metre. (1 mark)
2 A car manufacturer uses a mode
l to predict the fuel consumption, y miles per gallon (mpg),
for a specific model of car travelling at a speed of x mph.
y = β0.01x2 + 0.975x + 16, x . 0
a Use the model to find two speeds a
t which the car has a fuel consumption of
32.5 mpg. (3 marks)
b Rewrite
y in the form A β B(x β C)2, where A, B and C are constants to be found. (3 marks)
c Using your answ
er to part b, find the speed at which the car has the greatest fuel
efficiency. (1 mark)
d Use the model to calcula
te the fuel consumption of a car travelling at 120 mph.
Comment on the va
lidity of using this model for very high speeds. (2 marks)
3 A fertiliser company uses a model to deter
mine how the amount of fertiliser used, fΒ kilograms
per hectare, affects the grain yield g, measured in tonnes per hectare.
g = 6 + 0.03f β 0.000 06fΒ 2
a According to the model, how much grain would each hectare yield without any fertiliser?
(1 mark)
b One farmer currentl
y uses 20 kilograms of fertiliser per hectare. How much more fertiliser
would he need to use to increase his grain yield by 1 tonne per hectare? (4 marks)
4 A football stadium has 25 000 seats. The f
ootball club know from past experience that they will
sell only 10 000 tickets if each tick
et costs Β£30. They also expect to sell 1000 more tickets every
time the price goes down by Β£1.
a The number of tick
ets sold t can be modelled by the linear equation t = M β 1000p,
where Β£p is the price of each ticket and M is a constant. Find the value of M. (1 mark)E/P
E/P
E/P
E/PExercise 2H
For part a , make sure your
answer is in the context of
the model. Problem-solving
|
[
0.04503989219665527,
0.06857209652662277,
-0.0014795016031712294,
-0.04027184844017029,
-0.06593561172485352,
-0.05280234292149544,
-0.012304728850722313,
0.10046244412660599,
-0.023214293643832207,
-0.0041611636988818645,
0.0909179151058197,
-0.005397175904363394,
0.021432947367429733,
-0.03141961991786957,
-0.08564773947000504,
0.0221549104899168,
-0.04010157659649849,
0.04510694742202759,
-0.07586805522441864,
0.03479834645986557,
0.06793733686208725,
0.007757183630019426,
-0.034069035202264786,
-0.03932981193065643,
0.04728255793452263,
-0.019853539764881134,
-0.02370559424161911,
0.05443898215889931,
0.0504729263484478,
0.009580648504197598,
-0.06064196303486824,
0.027411293238401413,
0.017312197014689445,
0.045180805027484894,
0.0273716039955616,
0.07651840150356293,
0.06846954673528671,
-0.006026190239936113,
0.01758068986237049,
-0.02968171052634716,
-0.04780546575784683,
0.05502974987030029,
-0.11064121872186661,
0.008543020114302635,
0.028215551748871803,
0.044561147689819336,
-0.12053575366735458,
-0.0575273372232914,
0.056087836623191833,
-0.03121514990925789,
0.024624928832054138,
0.05688866227865219,
-0.05897108465433121,
-0.02740204706788063,
-0.0344557985663414,
0.033240269869565964,
-0.001383646740578115,
0.021336503326892853,
0.02314041368663311,
0.024450179189443588,
0.03997824713587761,
0.09260667860507965,
0.019462408497929573,
0.031019149348139763,
0.028305010870099068,
0.025477474555373192,
-0.0877380520105362,
-0.0006203870289027691,
0.012179636396467686,
0.055431123822927475,
-0.12360899895429611,
-0.0026028642896562815,
-0.008018179796636105,
-0.09220874309539795,
0.06175285205245018,
-0.03401106595993042,
0.013509917072951794,
0.04631989449262619,
-0.008806237950921059,
-0.014830009080469608,
0.008723522536456585,
-0.01107490248978138,
-0.05159676447510719,
0.03100472129881382,
0.03261149302124977,
0.015624293126165867,
-0.02596447989344597,
0.018921123817563057,
-0.02134343795478344,
-0.02790287882089615,
0.048169273883104324,
-0.0041326843202114105,
-0.057513389736413956,
-0.023944498971104622,
-0.009394613094627857,
-0.05214230343699455,
0.03129987418651581,
-0.15017761290073395,
0.10448996722698212,
0.14072275161743164,
-0.014391937293112278,
0.07837517559528351,
0.029137548059225082,
0.11540699005126953,
-0.009769409894943237,
0.03680975362658501,
-0.024318987503647804,
0.0641118511557579,
0.0021893351804465055,
-0.01933201774954796,
-0.03333837166428566,
-0.06229999661445618,
-0.004149626009166241,
0.03130190074443817,
0.09733950346708298,
-0.03212584927678108,
-0.012679800391197205,
-0.07742658257484436,
-0.06806638091802597,
-0.04032900929450989,
0.026418637484312057,
0.037343353033065796,
0.034576836973428726,
0.0388333685696125,
-0.043773919343948364,
-0.04231371358036995,
0.07397260516881943,
-0.005375723820179701,
-0.04864876717329025,
-0.019748574122786522,
-0.05891707167029381,
-0.02204572595655918,
0.020661499351263046,
-0.08779854327440262,
-0.025739414617419243,
0.02138868160545826,
0.02844100445508957,
0.05943400412797928,
-0.06350168585777283,
-0.05184066668152809,
0.00810934603214264,
0.009511054493486881,
0.03056473098695278,
-0.08766452223062515,
0.034307822585105896,
-0.004574257880449295,
-0.12131143361330032,
-0.024951374158263206,
0.016185447573661804,
0.015826979652047157,
0.0823490247130394,
0.004164750222116709,
0.05144655331969261,
-0.008738111704587936,
0.08222328871488571,
-0.04093138873577118,
-0.02797458879649639,
-0.006135032046586275,
-0.08004530519247055,
-0.021195849403738976,
-0.11220560222864151,
0.0012156739830970764,
-0.07232402265071869,
0.02021503634750843,
0.11284179240465164,
0.0057528321631252766,
0.007285742554813623,
0.07210511714220047,
-0.019657885655760765,
-0.06308819353580475,
0.00419017905369401,
0.014494509436190128,
-0.011077436618506908,
0.023215657100081444,
0.035147763788700104,
-0.010069615207612514,
0.029275523498654366,
0.07241038978099823,
-0.005263013299554586,
-0.014013133943080902,
-0.008011860772967339,
-0.07272147387266159,
0.002274187747389078,
-0.05437220633029938,
-0.025981102138757706,
-0.026714567095041275,
-0.03152170777320862,
0.017337100580334663,
-0.07608737796545029,
0.05262342095375061,
0.010424233041703701,
0.00371486390940845,
0.025524800643324852,
0.01750815100967884,
-0.12228530645370483,
-0.005800244398415089,
0.0545642152428627,
0.019268419593572617,
-0.08609327673912048,
-0.010747207328677177,
0.045229289680719376,
0.04187476634979248,
-0.022143185138702393,
-0.001786759472452104,
-0.06251779943704605,
0.018251650035381317,
0.015083389356732368,
-0.029622353613376617,
-0.0952502191066742,
0.13971702754497528,
-0.022588146850466728,
0.01410841103643179,
-0.016349269077181816,
-0.10750088095664978,
0.049935538321733475,
-0.05008526146411896,
-0.006331775337457657,
-0.052129682153463364,
-0.04881049692630768,
0.026535138487815857,
-0.03244052454829216,
-0.03604764863848686,
0.068553127348423,
8.283334736104305e-33,
-0.026714695617556572,
0.10658302903175354,
-0.03500398248434067,
-0.11749529093503952,
0.043269526213407516,
-0.013849013485014439,
0.010569627396762371,
-0.0061779688112437725,
0.07985454797744751,
0.03449637442827225,
0.08136659860610962,
0.03540608286857605,
-0.08269938826560974,
-0.01002670917659998,
-0.02537345699965954,
-0.0813581719994545,
0.09273187816143036,
-0.04670775309205055,
-0.06008906662464142,
0.05798628181219101,
0.006376693956553936,
-0.012157998979091644,
-0.02115587331354618,
0.041401151567697525,
-0.020238516852259636,
0.0008731968118809164,
0.06039763242006302,
-0.04110602289438248,
-0.0030643376521766186,
-0.021267380565404892,
-0.08235510438680649,
-0.05998583137989044,
0.0620480552315712,
0.09828364849090576,
-0.005409460514783859,
-0.06676137447357178,
0.05640033259987831,
0.016343768686056137,
-0.051246535032987595,
-0.08575870841741562,
0.06385911256074905,
0.016476888209581375,
0.027969613671302795,
-0.04112767055630684,
0.002941193524748087,
0.023632463067770004,
0.04976664483547211,
-0.03778611496090889,
-0.03726297244429588,
-0.00036226693191565573,
-0.0016224196879193187,
-0.022130049765110016,
0.07344266772270203,
-0.033292364329099655,
0.1106208860874176,
-0.014579051174223423,
-0.0207054540514946,
-0.06359806656837463,
-0.016000138595700264,
0.01898370124399662,
0.053638800978660583,
-0.06567930430173874,
0.009795751422643661,
0.006586379371583462,
-0.015692755579948425,
0.012384088709950447,
-0.01895092986524105,
-0.0746372789144516,
-0.05525313690304756,
0.01933320425450802,
-0.032679833471775055,
0.05752750486135483,
0.028765356168150902,
-0.020870257169008255,
0.009115169756114483,
-0.08820420503616333,
0.018925368785858154,
0.09238393604755402,
0.10152535885572433,
-0.03521095961332321,
-0.03168496862053871,
-0.03605867922306061,
0.07110924273729324,
-0.011559024453163147,
-0.07484903931617737,
-0.05905186012387276,
0.023854941129684448,
0.0383220911026001,
0.16586290299892426,
0.07189919054508209,
0.03560935705900192,
0.016100088134407997,
-0.059998489916324615,
-0.014947589486837387,
-0.033855337649583817,
7.703780315508011e-32,
-0.06993120163679123,
0.004871110897511244,
-0.054343849420547485,
-0.011821790598332882,
-0.036032527685165405,
0.07327637076377869,
0.001252829795703292,
-0.059077925980091095,
-0.05424446985125542,
-0.0009369880426675081,
0.07008089125156403,
0.03620341047644615,
-0.03557143360376358,
0.04588469862937927,
-0.10374606400728226,
-0.023163599893450737,
-0.058007556945085526,
0.017245832830667496,
-0.05318298563361168,
-0.01975671388208866,
0.007177742663770914,
0.05641219764947891,
0.006670263130217791,
0.055676333606243134,
0.046550873667001724,
-0.024182582274079323,
-0.0935545563697815,
-0.006141066085547209,
0.0024750938173383474,
-0.11318372935056686,
0.08339832723140717,
0.02692224085330963,
-0.050259366631507874,
-0.025977037847042084,
0.031211793422698975,
0.044199854135513306,
-0.02599341981112957,
0.04086396470665932,
-0.0271163173019886,
0.07805792987346649,
-0.052138544619083405,
-0.013448333367705345,
-0.029057452455163002,
-0.0052392794750630856,
0.027330495417118073,
0.055394239723682404,
0.0029585545416921377,
-0.037255167961120605,
0.0589807853102684,
0.027485709637403488,
-0.05979803204536438,
0.049469444900751114,
0.08165986090898514,
-0.0091962656006217,
0.05443422868847847,
-0.013173528015613556,
-0.029626060277223587,
-0.0003433329111430794,
0.009562538005411625,
-0.049722444266080856,
-0.03753766417503357,
0.05956588685512543,
-0.048116471618413925,
-0.037112556397914886
] |
35Quadratics
The total revenue, Β£r, can be calculated by multiplying the number of tickets sold by the price of
each ticket. This can be written as r = p(M β 1000p).
b Rearrange r into the f
orm A β B(p β C)2, where A, B and C are constants to be found. (3 marks)
c Using your answ
er to part b or otherwise, work out how much the football club should
charge for each ticket if they want to make the maximum amount of money. (2 marks)
1 Solve the follo
wing equations without a calculator. Leave your answers in surd form whereΒ necessary.
a y2 + 3y + 2 = 0 b 3x2 + 13x β 10 = 0 c 5x2 β 10x = 4x + 3 d (2x β 5)2 = 7
2 Sketch gra
phs of the following equations:
a y =
x2 + 5x + 4 b y = 2x2 + x β 3 c y = 6 β 10x β 4x2 d y = 15x β 2x2
3 f(x) = x2 + 3x β 5 and g(x) = 4x + k, where k is a constant.
a Given tha
t f(3) = g(3), find the value of k. (3 marks)
b Find the values of
x for which f(x) = g(x). (3 marks)
4 Solve the follo
wing equations, giving your answers correct to 3 significant figures:
a k2 + 11k β 1 = 0 b 2t2 β 5t + 1 = 0 c 10 β x β x2 = 7 d (3x β 1)2 = 3 β x2
5 Write each of these expressions in the form p (x + q)2 + r, where p , q and r are constants to beΒ found:
a x2 + 12x β 9 b 5x2 β 40x + 13 c 8x β 2x2 d 3x2 β (x + 1)2
6 Find the value k for which the equation 5x2 β 2x + k = 0 has exactly one solution. (2 marks)E
EMixed exercise 2Accident investigators are studying the stopping distance
of a particular car. When the car is travelling at 20
mph, its stopping distance
is 6 f
eet.
When the car is travelling at 30 mph, its stopping distance
is 14 f
eet.
When the car is travelling at 40 mph, its stopping distance
is 24 f
eet.
The investigators suggest that the stopping distance in feet, d, is a quadratic function of the speed in miles per hour, s.
a
Given that d(s
) = as2 + bs + c, find the values of the
constants a, b and c.
b At an accident sc
ene a car has left behind a skid that is
20 feet long.
Use your model to calculate the speed that this car was going at before the accident.Challenge Start by setting up three
si
multaneous equations. Combine
two different pairs of equations to
eliminate c . Use the results to find
the values of a and b first.Hint
|
[
0.09022817015647888,
0.0010313397506251931,
-0.054084960371255875,
-0.018644966185092926,
0.06131434440612793,
-0.015799805521965027,
0.04708779230713844,
0.03451910614967346,
-0.022265087813138962,
0.025579236447811127,
-0.054870445281267166,
-0.07496178895235062,
-0.07548219710588455,
0.0725722461938858,
0.029148831963539124,
-0.039537057280540466,
-0.023244379088282585,
0.005105598829686642,
-0.029462382197380066,
-0.0523245632648468,
-0.01384359784424305,
-0.10393975675106049,
-0.040841929614543915,
-0.013625174760818481,
-0.010811247862875462,
-0.012008007615804672,
-0.01743163727223873,
0.045981645584106445,
-0.043239425867795944,
-0.030125683173537254,
0.00895842257887125,
-0.07822143286466599,
0.07079987227916718,
-0.01785873807966709,
0.04517371952533722,
0.025690868496894836,
0.0369829498231411,
-0.032817237079143524,
0.012507057748734951,
0.053948890417814255,
-0.014298190362751484,
-0.016868464648723602,
-0.03742512688040733,
-0.03621603548526764,
0.029207082465291023,
0.0008486841688863933,
0.004855170845985413,
-0.042158808559179306,
0.04276059940457344,
0.053600113838911057,
0.13607759773731232,
0.09130170196294785,
-0.05400693044066429,
0.04061164706945419,
-0.020288629457354546,
-0.02362479642033577,
0.023041462525725365,
-0.010552008636295795,
-0.05198994651436806,
0.015520990826189518,
-0.09465932101011276,
0.03567156195640564,
0.04636437073349953,
-0.004166968632489443,
-0.046840839087963104,
0.020820867270231247,
-0.06852937489748001,
0.0442507304251194,
-0.05938608571887016,
0.06223900988698006,
-0.07375280559062958,
-0.030528321862220764,
0.010794744826853275,
-0.07357306033372879,
0.01494661532342434,
0.09840835630893707,
-0.04845519736409187,
0.010604474693536758,
-0.04719071090221405,
-0.03894485905766487,
-0.06014122813940048,
0.016681605949997902,
-0.0787653774023056,
-0.006043320521712303,
0.08701134473085403,
-0.016660042107105255,
0.12400704622268677,
0.04728377237915993,
0.030384985730051994,
0.011488670483231544,
0.013083234429359436,
-0.04691993072628975,
-0.034528929740190506,
0.009119529277086258,
-0.042346805334091187,
0.059189796447753906,
-0.0006556521402671933,
-0.07341137528419495,
0.038295961916446686,
0.11688133329153061,
0.005455945152789354,
0.02694215439260006,
-0.0065118130296468735,
-0.034757498651742935,
-0.02536621317267418,
-0.0566442646086216,
0.00011163684393977746,
0.11128979921340942,
0.023439807817339897,
-0.08256225287914276,
-0.005784697365015745,
0.024121373891830444,
0.04300211742520332,
0.042246732860803604,
-0.03747665882110596,
0.08238615840673447,
-0.005049436818808317,
-0.04578002169728279,
0.0014110172633081675,
-0.0250164233148098,
0.041463546454906464,
0.045131050050258636,
-0.00014661834575235844,
0.02455281838774681,
-0.08622605353593826,
0.05532741919159889,
0.033936403691768646,
0.021158017218112946,
-0.11830584704875946,
0.024921145290136337,
0.027555126696825027,
-0.05844493955373764,
-0.12488210201263428,
0.08880676329135895,
-0.006584756076335907,
0.06015560030937195,
0.013715428300201893,
0.024361934512853622,
0.03187042102217674,
-0.009261837229132652,
0.020684940740466118,
-0.013538628816604614,
0.0625024139881134,
-0.07369668036699295,
-0.050485145300626755,
-0.019263816997408867,
-0.043441105633974075,
-0.00609582057222724,
-0.05648711323738098,
0.024661609902977943,
-0.0024599097669124603,
0.01634964719414711,
0.021363435313105583,
0.04371250420808792,
-0.09447108954191208,
-0.006715252995491028,
0.0763908252120018,
-0.04076100513339043,
0.016227198764681816,
-0.011304318904876709,
-0.07992169260978699,
0.006450898014008999,
0.022140003740787506,
-0.014335842803120613,
0.005770035088062286,
0.04970625787973404,
0.037347037345170975,
-0.003938125912100077,
-0.13197779655456543,
0.015606258995831013,
0.06713370233774185,
-0.020661549642682076,
-0.13020844757556915,
0.11653278023004532,
-0.004706783685833216,
-0.016154009848833084,
0.025134067982435226,
0.09060576558113098,
0.012614528648555279,
-0.07844164222478867,
0.0361006036400795,
-0.04766806587576866,
0.06336396187543869,
-0.06195949763059616,
-0.035181574523448944,
0.0010836446890607476,
-0.008817672729492188,
0.0092930244281888,
-0.06261986494064331,
-0.0003257417993154377,
0.0751262903213501,
-0.014317606575787067,
-0.049106329679489136,
0.0005759265623055398,
0.028008360415697098,
0.000017088654203689657,
0.031730592250823975,
-0.050263531506061554,
-0.0011040755780413747,
0.042346078902482986,
0.04448635131120682,
-0.009729424491524696,
-0.0035826549865305424,
-0.0017966292798519135,
0.015513388440012932,
0.034151867032051086,
-0.048651088029146194,
-0.018470916897058487,
-0.006670660804957151,
0.03841845318675041,
-0.030666524544358253,
-0.061352964490652084,
0.05206575617194176,
0.021235141903162003,
0.06757082045078278,
-0.02186981588602066,
-0.029241764917969704,
-0.07276702672243118,
-0.10101917386054993,
0.04317427799105644,
-0.11138472706079483,
-0.006619274150580168,
0.021525854244828224,
5.4975887141241374e-33,
-0.020522093400359154,
0.10033714771270752,
-0.015729853883385658,
-0.052209991961717606,
-0.00492080207914114,
0.00046717465738765895,
0.04520829766988754,
-0.07147989422082901,
0.06809230148792267,
0.06250350177288055,
-0.025974128395318985,
0.06564003229141235,
0.0010420052567496896,
-0.012404029257595539,
-0.03980807214975357,
-0.09047123044729233,
0.03993350639939308,
0.021929671987891197,
-0.028873763978481293,
0.03301840275526047,
-0.04739133641123772,
7.796152772243659e-7,
-0.0035662311129271984,
0.012296821922063828,
-0.026508711278438568,
0.050762537866830826,
0.04624810442328453,
-0.05775056779384613,
-0.018593044951558113,
0.043808627873659134,
0.03648802638053894,
-0.09931696206331253,
0.040712080895900726,
0.008791157975792885,
-0.037289638072252274,
-0.06365276873111725,
0.08935786783695221,
0.04953770712018013,
-0.06498781591653824,
0.022733980789780617,
0.060016877949237823,
0.007679858710616827,
0.012781633995473385,
-0.05619211867451668,
0.10816796123981476,
-0.022523557767271996,
-0.04757075756788254,
-0.024186622351408005,
0.0779482051730156,
0.03127945214509964,
0.035575851798057556,
0.005898242350667715,
-0.005402781069278717,
0.05807237699627876,
-0.034193601459264755,
0.0058634295128285885,
0.003987216856330633,
-0.08596040308475494,
0.02714993990957737,
0.05466888099908829,
0.05228513479232788,
-0.012709406204521656,
-0.025191910564899445,
0.06965773552656174,
-0.08102357387542725,
0.10450229048728943,
-0.042565975338220596,
0.0013086412800475955,
0.042358268052339554,
0.02798207476735115,
-0.13176444172859192,
0.027952278032898903,
-0.020097678527235985,
-0.06228446587920189,
-0.09859038144350052,
0.023458389565348625,
-0.006654373370110989,
0.07339946180582047,
0.052719730883836746,
0.01319680456072092,
-0.043864376842975616,
-0.08282644301652908,
0.09991498291492462,
-0.06650226563215256,
-0.024633055552840233,
0.011381682939827442,
0.06708889454603195,
0.02570277266204357,
-0.0009503333130851388,
0.04988693818449974,
-0.001539830002002418,
-0.08288571238517761,
0.09889185428619385,
-0.030345363542437553,
0.0006325109279714525,
7.058470944978626e-32,
-0.09013915807008743,
0.0029985336586833,
-0.01524706557393074,
0.08367682248353958,
0.015072239562869072,
-0.016965216025710106,
0.026766575872898102,
-0.058824483305215836,
-0.001417881459929049,
-0.07061657309532166,
0.01782356947660446,
-0.024077890440821648,
-0.0448048934340477,
0.03464338928461075,
-0.11158543825149536,
-0.02930157259106636,
-0.11246820539236069,
0.038050975650548935,
-0.025246847420930862,
-0.017870571464300156,
0.03173724561929703,
0.07575476169586182,
-0.10185660421848297,
0.0188643466681242,
0.03870898857712746,
0.04540141299366951,
-0.017418239265680313,
-0.025161784142255783,
-0.037692759186029434,
-0.06710807979106903,
-0.029541965574026108,
-0.0722331553697586,
0.020108822733163834,
-0.04010801389813423,
0.07999135553836823,
-0.01682012714445591,
0.03922600671648979,
-0.016495754942297935,
-0.016620859503746033,
0.06341850757598877,
0.027591312304139137,
-0.07640748471021652,
-0.034562285989522934,
-0.010829247534275055,
0.09332412481307983,
0.05010829120874405,
-0.07101048529148102,
-0.023562069982290268,
0.013755091466009617,
-0.048954591155052185,
0.08554955571889877,
0.05395912006497383,
-0.03009255789220333,
0.04135514050722122,
0.055061038583517075,
-0.04368063807487488,
-0.08141470700502396,
0.0433269627392292,
0.055885907262563705,
0.011560444720089436,
-0.028927644714713097,
0.11418171972036362,
-0.036460891366004944,
-0.01781601272523403
] |
36
Chapter 2
7 Given tha
t for all values of x:
3x2 + 12x + 5 = p(x + q)2 + r
a find the values of
p, q and r. (3 marks)
b Hence solve the equation 3
x2 + 12x + 5 = 0. (2 marks)
8 The function f is defined as f(x
) = 22x β 20(2x) + 64, x β β .
a Write f(x
) in the form (2x β a)(2x β b), where a and b are real constants. (2 marks)
b Hence find the two roots of
f(x). (2 marks)
9 Find, as surds, the r
oots of the equation:
2(x + 1)(x β 4) β (x β 2)2 = 0.
10 Use algebr
a to solve (x β 1)(x + 2) = 18.
11 A diver launches herse
lf off a springboard. The height of the diver, in metres, above the pool
tΒ seconds after launch can be modelled by the following function:
h(t) = 5t β 10t2 + 10, t > 0
a How high is the springboard a
bove the water? (1 mark)
b Use the model to find the time at w
hich the diver hits the water. (3 marks)
c Rearrange h(
t) into the form A β B(t β C)2 and give the values of the constants
A, B and C. (3 marks)
d Using your answ
er to part c or otherwise, find the maximum height of the diver, and
the time at which this maximum height is reached. (2 marks)
12 For this question, f(x
) = 4kx2 + (4k + 2)x + 1, where k is a real constant.
a Find the discriminant of f(x
) in terms of k. (3 marks)
b By simplifying your answ
er to part a or otherwise, prove that f(x) has two distinct
real roots for all non-zero values of k. (2 marks)
c Explain why f(
x) cannot have two distinct real roots when k = 0. (1 mark)
13 Find all of the r
oots of the function r(x) = x8 β 17x4 + 16. (5 marks)
14 Lynn is selling cushions as part of
an enterprise project. On her first attempt, she sold 80
cushions at the cost of Β£15 each. She hopes to sell more cushions next time. Her adviser
suggests that she can expect to sell 10 more cushions for every Β£1 that she lowers the price.
a The number of cushions sold
c can be modelled by the equation c = 230 β Hp, where
Β£p is the price of each cushion and H is a constant. Determine the value of H. (1 mark)
To model her tota
l revenue, Β£r, Lynn multiplies the number of cushions sold by the price of
each cushion. She writes this as r = p(230 β Hp).b
Rearrange
r into the form A β B( p β
C )2, where A, B and C are constants to be
found. (3 marks)
c Using your answ
er to part b or otherwise, show that Lynn can increase her revenue by Β£122.50
through lowering her prices, and state the optimum selling price of a cushion. (2 marks)E
E/P
E/P
E/P
E/P
E/P
|
[
-0.0035991687327623367,
0.11988818645477295,
0.015049285255372524,
0.03419390320777893,
0.004889210686087608,
0.05366562306880951,
0.0025761560536921024,
0.076433464884758,
-0.053402941673994064,
0.08001580834388733,
0.0011862489627674222,
-0.0637127235531807,
-0.07562480121850967,
-0.07429938018321991,
0.013717255555093288,
-0.047452107071876526,
-0.042903877794742584,
0.002447668928653002,
-0.07105448096990585,
-0.019588623195886612,
0.002384357387199998,
-0.015055225230753422,
-0.05705072730779648,
-0.0021936146076768637,
0.09581272304058075,
-0.06516416370868683,
0.018515540286898613,
-0.03652295097708702,
0.005391386337578297,
-0.037832532078027725,
-0.03855659440159798,
-0.0295829139649868,
0.0449170358479023,
-0.02127617783844471,
0.07296554744243622,
0.041011691093444824,
-0.01750657707452774,
0.018062099814414978,
0.08347020298242569,
-0.044908661395311356,
-0.025425538420677185,
0.038412708789110184,
-0.054525233805179596,
-0.0945417508482933,
0.06780590116977692,
-0.08258352428674698,
-0.07143603265285492,
-0.034716252237558365,
0.04213656857609749,
-0.009074172005057335,
0.01043043565005064,
0.0010106812696903944,
-0.08712320029735565,
0.10684077441692352,
-0.004392622970044613,
-0.016853060573339462,
0.050368163734674454,
0.013543159700930119,
-0.03220713138580322,
0.01863979920744896,
-0.05815132334828377,
0.07275620847940445,
0.015072517096996307,
0.07039330899715424,
0.038216061890125275,
0.05712644010782242,
-0.06437965482473373,
-0.05883558467030525,
0.0031625533010810614,
0.020234709605574608,
-0.1357659250497818,
0.07604850828647614,
-0.0293258186429739,
-0.022733716294169426,
0.015644283965229988,
-0.07542545348405838,
-0.07379663735628128,
-0.0808507427573204,
0.007369036320596933,
0.003399485955014825,
-0.01117402222007513,
-0.03557189926505089,
0.0007553547038696706,
0.028418635949492455,
0.002557590138167143,
0.023288849741220474,
0.07627057284116745,
0.007422648835927248,
0.04080814868211746,
0.04520341008901596,
0.07482928782701492,
-0.017569223418831825,
0.018672989681363106,
-0.06566944718360901,
-0.033601272851228714,
-0.015957187861204147,
0.03415503725409508,
-0.03721870854496956,
0.01751197688281536,
0.15234559774398804,
0.04609307646751404,
-0.02670271135866642,
-0.006057312712073326,
0.041305724531412125,
-0.03704589605331421,
-0.049249663949012756,
0.025034921243786812,
-0.042508114129304886,
0.09529902786016464,
-0.0767715573310852,
-0.06630858778953552,
-0.08037140965461731,
0.013416483998298645,
0.01642535626888275,
0.0289935152977705,
-0.06076068431138992,
0.07199599593877792,
-0.032437872141599655,
-0.04564070701599121,
-0.043629854917526245,
-0.038371749222278595,
-0.04073077812790871,
0.036682967096567154,
-0.013018508441746235,
-0.05716883763670921,
-0.045862723141908646,
0.030296694487333298,
0.058534953743219376,
-0.007835529744625092,
-0.045987170189619064,
0.04624431952834129,
-0.03283695504069328,
-0.057037461549043655,
-0.05347256734967232,
0.04979991540312767,
0.03150323033332825,
-0.07897108048200607,
0.03739587590098381,
0.04468972980976105,
-0.05822925269603729,
-0.022726958617568016,
0.07850724458694458,
0.005576062947511673,
0.04001772403717041,
-0.00438532093539834,
0.025700001046061516,
-0.007881758734583855,
-0.05007738620042801,
-0.02962663397192955,
0.04655176401138306,
-0.0022354661487042904,
-0.008360396139323711,
-0.00727419089525938,
0.11261782050132751,
-0.003144612303003669,
-0.07005719840526581,
-0.01218713540583849,
-0.06717969477176666,
-0.02190341055393219,
0.01968620903789997,
-0.0835932269692421,
0.03924863785505295,
-0.0005187478382140398,
0.05766637995839119,
-0.018185298889875412,
0.001273467089049518,
0.07262197136878967,
0.06697437167167664,
-0.04028930887579918,
-0.012049628421664238,
0.022453229874372482,
0.05502942204475403,
0.012838282622396946,
0.11453066021203995,
-0.0248867180198431,
-0.027010276913642883,
0.09389780461788177,
-0.002097839256748557,
-0.018391434103250504,
0.0586601085960865,
-0.000874058052431792,
-0.02143121138215065,
0.00808333232998848,
0.026541458442807198,
-0.04498261213302612,
0.03573721647262573,
-0.022864598780870438,
0.009881851263344288,
0.023778239265084267,
-0.009041556157171726,
0.0782380998134613,
-0.04192134737968445,
0.027052609249949455,
0.04530870169401169,
-0.11754082888364792,
-0.08001725375652313,
0.004302320070564747,
0.08057067543268204,
-0.047367312014102936,
0.0029534187633544207,
0.05315960943698883,
-0.037372931838035583,
0.03632489964365959,
0.028634849935770035,
0.020845524966716766,
-0.04728913679718971,
-0.0702999085187912,
-0.035688504576683044,
0.010061189532279968,
-0.010639539919793606,
-0.030821092426776886,
0.018596086651086807,
-0.01941474713385105,
-0.03782527148723602,
0.012384398840367794,
-0.04579107463359833,
0.027498751878738403,
-0.05174179747700691,
-0.062231600284576416,
0.09147056937217712,
-0.0675756186246872,
-0.04843929782509804,
0.060296639800071716,
-5.748793326262799e-33,
-0.03842167928814888,
0.07321788370609283,
-0.12866033613681793,
-0.03292415291070938,
-0.05956622213125229,
-0.06151115894317627,
-0.03375420346856117,
-0.07689785957336426,
0.09950681030750275,
0.0459163673222065,
0.02171410247683525,
0.03679956495761871,
-0.034222621470689774,
0.057653918862342834,
-0.036583393812179565,
-0.06141732260584831,
0.021939612925052643,
0.020645393058657646,
-0.02021700330078602,
-0.04691630229353905,
-0.006366803776472807,
0.06062793731689453,
-0.03805135563015938,
0.017381785437464714,
-0.0051621911115944386,
0.03383190929889679,
0.05969942733645439,
-0.08205941319465637,
-0.006431394722312689,
0.06320951879024506,
-0.08033301681280136,
0.013610733672976494,
0.047735344618558884,
0.043032582849264145,
-0.061475399881601334,
-0.07357843965291977,
0.0639728456735611,
0.018368810415267944,
0.006135815754532814,
-0.014447204768657684,
0.05471653863787651,
0.040464505553245544,
-0.008643520064651966,
-0.034989021718502045,
0.011543826200067997,
0.020281372591853142,
0.09408607333898544,
0.0009821103885769844,
-0.020257441326975822,
0.08382442593574524,
-0.012945146299898624,
-0.05897888168692589,
0.04443897679448128,
0.027860315516591072,
0.056838855147361755,
0.005253149662166834,
0.0016534053720533848,
-0.03313826769590378,
0.04633161798119545,
-0.00047785398783162236,
-0.0138729028403759,
0.006564244627952576,
0.045603614300489426,
0.03801076114177704,
-0.05086241289973259,
-0.02291061170399189,
-0.006664537824690342,
-0.0399244949221611,
0.006347945891320705,
0.06726338714361191,
-0.14065149426460266,
-0.006213564891368151,
-0.039028726518154144,
-0.09563346952199936,
-0.053479112684726715,
0.09165503829717636,
-0.057974155992269516,
0.048398457467556,
-0.001874082488939166,
-0.06339171528816223,
-0.03234288468956947,
-0.01320157665759325,
0.08208080381155014,
-0.0853763297200203,
-0.159587562084198,
0.06409668177366257,
0.09280446171760559,
0.02960379794239998,
0.008609188720583916,
-0.0008818116621114314,
0.018122481182217598,
0.02603061869740486,
-0.01928824558854103,
-0.0737299993634224,
0.08936111629009247,
9.144661323938043e-32,
-0.013431712985038757,
-0.07722613215446472,
-0.07987487316131592,
-0.0029570998158305883,
0.09242761135101318,
0.06180492043495178,
0.023053515702486038,
-0.043976809829473495,
0.06392408162355423,
-0.01066124252974987,
0.01793779619038105,
-0.009099294431507587,
-0.037617821246385574,
0.02066662162542343,
-0.08395793288946152,
-0.03704598918557167,
-0.04561656713485718,
0.02569586969912052,
-0.014745148830115795,
-0.07461564242839813,
-0.024270664900541306,
0.008731972426176071,
-0.06542039662599564,
0.06441837549209595,
0.04507758095860481,
0.06520406901836395,
-0.033185139298439026,
-0.009632698260247707,
-0.038278788328170776,
-0.027642959728837013,
0.05265195295214653,
0.019566597416996956,
-0.002481437288224697,
-0.035738229751586914,
-0.024802248924970627,
0.0037243778351694345,
0.1323266476392746,
-0.00284598208963871,
-0.03518661856651306,
0.018327943980693817,
-0.010631588287651539,
-0.07637369632720947,
-0.052054837346076965,
0.03388485312461853,
-0.046450529247522354,
-0.05726179480552673,
0.00001196421726490371,
-0.05795327201485634,
0.08162092417478561,
-0.027331657707691193,
0.00746527872979641,
0.08070086687803268,
0.03684841841459274,
-0.0092125553637743,
0.024833373725414276,
-0.06602030247449875,
-0.08645738661289215,
0.04961720481514931,
-0.049543749541044235,
-0.10447067767381668,
-0.05445891618728638,
0.05403536558151245,
-0.011955969035625458,
0.009751548990607262
] |
37Quadratics
1 To solve a quadratic equation by factorising:
β Write the equation in the f
orm ax2 + bx + c = 0
β Factorise the l
eft-hand side
β Set each factor equal to z
ero and solve to find the value(s) of x
2 The solutions of the equation ax2 + bx + c = 0 where a β 0 are given by the formula:
x = βb Β± ββ―_______ b 2 β 4ac _____________ 2a
3 x2 + bx = (x + b __ 2 ) 2
β ( b __ 2 ) 2
4 ax2 + bx + c = a (x + b ___ 2a ) 2
+ (c β b2 ___ 4a2 )
5 The set of possibl
e inputs for a function is called the domain.
The set of possible outputs of a function is called the range.
6 The roots
of a function are the values of x for which f(x) = 0.
7 You can find the coor
dinates of a turning point of a quadratic graph by completing the
square. If f(x) = a(x + p)2 + q, the graph of y = f(x) has a turning point at (βp, q).
8 For the quadratic function f(
x) = ax2 + bx + c = 0, the expression b2 β 4ac is called the
discriminant. The value of the discriminant shows how many roots f(x) has:
β If b2 β 4ac . 0 then a quadratic function has two distinct real roots.
β If b2 β 4ac = 0 then a quadratic function has one repeated real root.
β If b2 β 4ac , 0 then a quadratic function has no real roots
9 Quadratics can be used to model r
eal-life situations.Summary of key pointsa The ratio of the lengths a : b in this line is the same as the r atio
of the lengths b : c.
a
b c
Show that this ratio is 1 + ββ―__
5 ______ 2 : 1.
b Show also that the infinite squar
e root
ββ―_______________________ 1 + ββ―___________________ 1 + ββ―______________ 1 + ββ―__________ 1 + ββ―______ 1 + β¦ = 1 + ββ―__
5 ______ 2 Challenge
|
[
-0.006938998121768236,
0.08979608118534088,
0.022742995992302895,
0.017774708569049835,
-0.014290140941739082,
0.06696460396051407,
0.01830935850739479,
0.025132732465863228,
-0.10102827101945877,
0.05561348795890808,
0.002535366453230381,
-0.09215235710144043,
0.011073657311499119,
0.014074234291911125,
0.07127033174037933,
0.0392443984746933,
-0.12417607009410858,
0.06832187622785568,
-0.040773842483758926,
-0.025904115289449692,
0.034630414098501205,
-0.10098899155855179,
-0.04588869959115982,
-0.009331054985523224,
0.011882205493748188,
-0.050474394112825394,
-0.0628901794552803,
-0.037845853716135025,
0.03405231609940529,
0.004721957724541426,
-0.017398657277226448,
0.10173409432172775,
0.06305167824029922,
-0.008137575350701809,
0.007729425095021725,
0.058824196457862854,
0.07091888040304184,
0.02664436213672161,
-0.03738073632121086,
-0.05579327791929245,
-0.014026558957993984,
0.0453752838075161,
-0.053625959903001785,
0.01935657672584057,
0.05269157141447067,
-0.09856463968753815,
-0.02676151879131794,
-0.04713461548089981,
0.06388187408447266,
-0.043223120272159576,
0.023556670174002647,
0.04979265108704567,
-0.09410693496465683,
0.005453691817820072,
-0.10803665220737457,
-0.030898062512278557,
-0.0575605183839798,
0.016727829352021217,
-0.04186217859387398,
0.039114490151405334,
0.012370901182293892,
0.015711063519120216,
0.07646841555833817,
0.0943429097533226,
0.02950398623943329,
0.03195743262767792,
-0.03241058066487312,
-0.013669084757566452,
-0.004690877161920071,
0.05502814054489136,
0.004012431483715773,
0.02380218915641308,
-0.009609390050172806,
0.017144186422228813,
0.030273618176579475,
0.03310618922114372,
-0.08073664456605911,
0.027627896517515182,
0.023219408467411995,
-0.01572561450302601,
0.06003633514046669,
-0.025507168844342232,
0.1025533378124237,
-0.002973612630739808,
-0.02915644459426403,
-0.013013336807489395,
0.10289936512708664,
0.04967191815376282,
-0.0010476680472493172,
0.001927930978126824,
0.026509517803788185,
-0.022338809445500374,
0.0195442084223032,
-0.029049329459667206,
-0.02243899740278721,
-0.08780954033136368,
0.06946712732315063,
0.026223331689834595,
0.030217718333005905,
0.003022825578227639,
0.01229229848831892,
-0.01683884486556053,
0.006677027326077223,
-0.022825511172413826,
-0.034550990909338,
-0.0398014634847641,
-0.054314132779836655,
0.01928088814020157,
0.09354989230632782,
-0.03182647004723549,
-0.14235681295394897,
-0.019782474264502525,
0.011267617344856262,
0.012133006006479263,
0.05237923935055733,
-0.09701794385910034,
0.09516332298517227,
-0.011669307947158813,
0.047093234956264496,
-0.040033984929323196,
-0.009174216538667679,
0.002397548407316208,
0.04463127627968788,
-0.1067296639084816,
0.06293616443872452,
-0.002709448104724288,
-0.004395700059831142,
0.027451610192656517,
-0.062237367033958435,
-0.03876524046063423,
0.03873080015182495,
-0.042098186910152435,
-0.1113705188035965,
-0.08706525713205338,
0.05675558000802994,
-0.006100914441049099,
-0.009171429090201855,
0.11978546530008316,
0.02145904302597046,
-0.05694139748811722,
0.06448837369680405,
0.02660505846142769,
0.013847201131284237,
-0.005767060909420252,
0.05776289477944374,
-0.10479355603456497,
-0.0244605615735054,
-0.09587588906288147,
-0.005109739489853382,
-0.018114613369107246,
-0.04095236212015152,
-0.02627716027200222,
-0.007943146862089634,
0.07891155034303665,
0.03810811787843704,
-0.08903549611568451,
0.0029327189549803734,
-0.09485211968421936,
-0.035868410021066666,
0.07573989778757095,
-0.04233148694038391,
-0.04278334975242615,
-0.05669171363115311,
0.03080475516617298,
0.006932633928954601,
0.03624683991074562,
-0.10873496532440186,
0.0048272451385855675,
0.01677604205906391,
0.049586351960897446,
0.002500369679182768,
0.04282066971063614,
-0.03080136887729168,
0.027014032006263733,
-0.05262598395347595,
-0.027406085282564163,
0.07915957272052765,
0.028319532051682472,
0.09949950873851776,
0.02345304563641548,
0.06780735403299332,
0.010045679286122322,
0.01947912387549877,
0.0021946681663393974,
-0.00582119170576334,
-0.010771660134196281,
0.02780657261610031,
-0.11396197974681854,
-0.026725929230451584,
-0.09016183018684387,
-0.028278015553951263,
0.03211300075054169,
-0.027569198980927467,
0.06741811335086823,
-0.05287974327802658,
-0.009917094372212887,
0.0015833412762731314,
-0.023896656930446625,
-0.08332252502441406,
0.08399797230958939,
0.11795932799577713,
-0.007471039891242981,
0.09670429676771164,
-0.0213047843426466,
0.0035403461661189795,
-0.027438877150416374,
-0.04673681780695915,
-0.01892133429646492,
-0.002911983523517847,
0.06378387659788132,
-0.05027450621128082,
-0.028225867077708244,
0.08517032116651535,
-0.045401863753795624,
-0.04073739051818848,
-0.0986407995223999,
-0.04555531218647957,
-0.04754534736275673,
-0.027453763410449028,
0.03640926629304886,
-0.024449249729514122,
0.019162733107805252,
0.044830240309238434,
-7.24877187237004e-33,
-0.01699289120733738,
-0.05905331298708916,
-0.10567774623632431,
0.0086777089163661,
-0.026793094351887703,
0.03332289680838585,
0.02204088680446148,
-0.11799082905054092,
0.12197649478912354,
0.026749780401587486,
0.06247217208147049,
0.003658342408016324,
-0.036149781197309494,
-0.016702977940440178,
-0.05088157579302788,
0.00007840660691726953,
-0.04592614248394966,
0.011263750493526459,
-0.00004648641333915293,
0.023781027644872665,
0.07821980118751526,
-0.013154697604477406,
-0.027774259448051453,
-0.08680085837841034,
0.0057838368229568005,
0.04117296263575554,
0.0698971077799797,
-0.03366474434733391,
0.019931018352508545,
0.023197732865810394,
-0.016938883811235428,
-0.011415286920964718,
0.07706550508737564,
-0.006407794542610645,
-0.013513906858861446,
-0.03153513744473457,
-0.04954967275261879,
0.017030971124768257,
-0.03296658396720886,
-0.04220585525035858,
0.02727748081088066,
0.03220124542713165,
0.04093760624527931,
0.004755454137921333,
0.005739424377679825,
-0.012110859155654907,
0.011570451781153679,
0.019755523651838303,
0.01975853368639946,
0.05423073098063469,
-0.03237694129347801,
-0.060537826269865036,
-0.021044576540589333,
0.018858157098293304,
-0.013687720522284508,
0.08681806176900864,
0.007933004759252071,
-0.040640998631715775,
-0.02687275968492031,
0.0015892512165009975,
0.023958010599017143,
-0.03738779202103615,
0.02408112958073616,
0.06436315178871155,
-0.005077493377029896,
0.10928894579410553,
0.04716792702674866,
0.04521888867020607,
0.005699914414435625,
0.046015553176403046,
-0.07209129631519318,
0.03843965008854866,
-0.08556462079286575,
0.07264987379312515,
-0.058067962527275085,
0.033431317657232285,
-0.05054726451635361,
0.08313644677400589,
-0.01529909297823906,
-0.04252786189317703,
-0.04388783127069473,
0.05412482097744942,
-0.026592670008540154,
-0.02240120619535446,
-0.08261707425117493,
-0.03746785596013069,
0.05556599050760269,
-0.0009990056278184056,
0.025921182706952095,
-0.08791342377662659,
0.024750597774982452,
-0.014466412365436554,
-0.004443461541086435,
-0.0015446055913344026,
0.07605805993080139,
8.151867594845134e-32,
0.000057209821534343064,
-0.04708690196275711,
-0.03843427821993828,
0.01570395566523075,
0.032794538885354996,
0.02983097918331623,
-0.05002281814813614,
-0.005449015647172928,
-0.016941383481025696,
-0.08432881534099579,
0.008708410896360874,
0.09526871144771576,
-0.040396515280008316,
0.027851462364196777,
-0.07463227957487106,
0.014557944610714912,
0.008311889134347439,
0.053500622510910034,
-0.02839083969593048,
0.006605868693441153,
-0.038858797401189804,
0.03708809241652489,
-0.05800798535346985,
0.039709240198135376,
0.0178512129932642,
0.04525944963097572,
0.004130448680371046,
-0.020850636065006256,
0.050960320979356766,
-0.032849449664354324,
0.11831249296665192,
0.04084768518805504,
-0.061720822006464005,
-0.07859878987073898,
0.05336272343993187,
0.04224739968776703,
0.06796643882989883,
-0.011809106916189194,
-0.0495881661772728,
0.006737350486218929,
-0.012437011115252972,
-0.04421449452638626,
-0.10650072246789932,
-0.02876313403248787,
-0.07223159819841385,
-0.0675911083817482,
-0.03963626176118851,
-0.11463416367769241,
0.017834559082984924,
0.0015546850627288222,
-0.03450298681855202,
-0.02255047671496868,
0.024714602157473564,
0.04681514576077461,
0.05688639357686043,
-0.05019037052989006,
-0.06857313215732574,
-0.012796551920473576,
0.07382268458604813,
-0.036606647074222565,
-0.10115360468626022,
0.08607721328735352,
-0.05825045332312584,
-0.0716017633676529
] |
38
Equations and
inequalities
After completing this chapter you should be able to:
β Solve linear simultaneous equations using elimination or
substit
ution β pages 39 β 40
β Solve simultaneous equations: one linear and one quadratic
β pages 41 β 42
β Interpret algebraic solutions of equations graphically β pages 42 β 45
β Solve linear inequalities β pages 46 β 48
β Solve quadratic inequalities β pages 48 β 51
β Interpret inequalities graphically β pages 51 β 53
β Represent linear and quadratic inequalities graphically β pages 53 β 55Objectives
1 A = {factors of 12}
B = {factors of 20}
Write do
wn the
numbers in each of these sets:
a A β©
B b (A
βͺ B)9
β GCSE Mathematics
2 Simplify these expressions.
a ββ―___
75 b 2 ββ―___
45 + 3 ββ―___
32 ___________ 6
β Section 1.5
3 Match the equations to the correct graph. Label
the points of int
ersection with the axes and the
coordinates of the turning point.
a y =
9 β x2 b y = (x β 2)2 + 4
c y =
(x β 7)(2x + 5)
y
xii i
y
xiii
y
xO O
O
β Section 2.47AB
9
116
1231
25
10
20
j 134Prior knowledge check
Food scientists use regions on graphs to optimise athletesβ nutritional intake and ensure they satisfy the minimum dietary requirements for calories and vitamins.3
|
[
0.0439019538462162,
0.07101308554410934,
-0.0009480773587711155,
-0.018568065017461777,
-0.04018965736031532,
0.065226711332798,
-0.03479668125510216,
0.04303557798266411,
-0.13567879796028137,
0.06996122747659683,
0.012954873964190483,
-0.04754377529025078,
0.02475557290017605,
0.022769248113036156,
0.037237878888845444,
0.07559607923030853,
-0.08398901671171188,
-0.02117026224732399,
0.006215506233274937,
-0.04556576907634735,
0.017698636278510094,
-0.09778348356485367,
-0.07326120138168335,
0.012298007495701313,
0.04801858589053154,
-0.06596511602401733,
-0.08845988661050797,
-0.0009358801762573421,
0.02922229655086994,
-0.02226717211306095,
-0.008794021792709827,
-0.002748088911175728,
0.1013459712266922,
-0.0028272117488086224,
-0.0027361037209630013,
0.020969009026885033,
0.07802645117044449,
0.0020581737626343966,
0.04197167605161667,
-0.06128070876002312,
-0.04803285375237465,
0.023102132603526115,
-0.09971176087856293,
0.06220627948641777,
-0.006181558594107628,
-0.017104148864746094,
0.003007110906764865,
-0.04963938146829605,
0.05535869672894478,
-0.08047152310609818,
-0.011310611851513386,
-0.033997662365436554,
0.04862665385007858,
0.07372739166021347,
-0.06415904313325882,
-0.11065226048231125,
-0.034505922347307205,
0.010431326925754547,
-0.0773966908454895,
-0.0010668020695447922,
0.006907028146088123,
-0.0442044623196125,
0.018966736271977425,
0.051946837455034256,
0.08305001258850098,
0.07561936229467392,
0.06205231323838234,
0.06850501149892807,
-0.013721322640776634,
0.07158119231462479,
-0.13340750336647034,
-0.014979255385696888,
-0.03828456252813339,
-0.019676076248288155,
0.03164232149720192,
0.05319194868206978,
-0.03645973280072212,
-0.03812360391020775,
-0.004541925620287657,
0.0043778494000434875,
0.03750113397836685,
0.0546254999935627,
0.04847247153520584,
-0.010573641397058964,
-0.01825074665248394,
-0.0922444760799408,
0.06647848337888718,
0.05131160840392113,
-0.020966511219739914,
-0.0413958877325058,
0.011027084663510323,
-0.01962849497795105,
0.054582126438617706,
-0.09350953251123428,
-0.011921070516109467,
-0.11921355128288269,
0.040242090821266174,
-0.014341389760375023,
0.0448499396443367,
0.02127411961555481,
0.0011609811335802078,
-0.1220351904630661,
0.11372482031583786,
-0.020914794877171516,
-0.03141343221068382,
0.03684777766466141,
0.045266151428222656,
-0.07980582863092422,
0.01802186109125614,
-0.04472508653998375,
-0.03652992472052574,
-0.042361993342638016,
0.007766027934849262,
0.020078103989362717,
0.03235212340950966,
-0.05458592250943184,
0.032242681831121445,
-0.009141677059233189,
0.056319709867239,
-0.07539749145507812,
-0.09769041836261749,
-0.01968490146100521,
0.09363541007041931,
-0.02333819679915905,
-0.0001074472238542512,
0.022616034373641014,
0.01613132655620575,
0.012634765356779099,
-0.0676485076546669,
-0.08768026530742645,
0.03049270063638687,
0.01281412597745657,
-0.05277201905846596,
-0.027967561036348343,
0.047533340752124786,
-0.01858953945338726,
-0.043390993028879166,
0.035639792680740356,
-0.01069659274071455,
-0.05244581401348114,
-0.04844992607831955,
0.06961893290281296,
0.07202162593603134,
0.007441242691129446,
0.041582927107810974,
-0.008300220593810081,
-0.005448037292808294,
0.05179379880428314,
-0.043526507914066315,
0.0022341678850352764,
-0.057681623846292496,
0.0211909431964159,
0.03535780310630798,
0.1114027202129364,
-0.08704648911952972,
0.0019858817104250193,
-0.04600276052951813,
0.013377063907682896,
-0.0069948770105838776,
0.04365799203515053,
-0.06373584270477295,
0.0036639217287302017,
0.023851359263062477,
0.020394014194607735,
0.010115095414221287,
0.010120350867509842,
-0.03081716038286686,
0.01611252874135971,
-0.04762192443013191,
-0.05078009516000748,
0.07478141039609909,
-0.004347455687820911,
0.028557188808918,
0.09521669894456863,
-0.027102578431367874,
0.013160351663827896,
0.09594568610191345,
-0.01744425855576992,
-0.08410994708538055,
0.08869680762290955,
-0.0007267125183716416,
-0.06262871623039246,
0.035914771258831024,
-0.046823885291814804,
0.0005354997701942921,
-0.009651265107095242,
0.005852640140801668,
0.000019341156075824983,
-0.004614145494997501,
-0.010600920766592026,
-0.004878406412899494,
-0.0686187669634819,
0.0157197043299675,
0.07118359953165054,
-0.13780651986598969,
-0.0826997458934784,
0.002441862365230918,
-0.0058210245333611965,
-0.08759912848472595,
-0.022999398410320282,
0.009512958116829395,
-0.0486455112695694,
-0.020695896819233894,
0.029880167916417122,
0.014705777168273926,
-0.03574775159358978,
-0.04318568482995033,
-0.04963604733347893,
0.0371062308549881,
0.02686464600265026,
0.00808032788336277,
0.035348664969205856,
-0.012632239609956741,
-0.014901612885296345,
0.016676612198352814,
-0.07890928536653519,
0.07944370806217194,
-0.059312909841537476,
-0.01519203744828701,
0.03102259710431099,
-0.10207077860832214,
-0.0014680427266284823,
0.1062433198094368,
-3.1198942501968186e-33,
-0.00885721854865551,
0.007362757809460163,
-0.12963145971298218,
-0.07424346357584,
-0.022979333996772766,
0.05036032572388649,
0.057933419942855835,
-0.10117983818054199,
0.051790833473205566,
-0.04510687291622162,
0.07139981538057327,
-0.002677799668163061,
-0.07469931989908218,
0.030623318627476692,
-0.05968025326728821,
-0.09867655485868454,
0.047489508986473083,
-0.07760906219482422,
0.05529510974884033,
0.017530066892504692,
0.02348063327372074,
0.038916271179914474,
0.020987384021282196,
-0.05450280383229256,
-0.03799933195114136,
0.03263922408223152,
0.10960830003023148,
0.00905810110270977,
0.006262657232582569,
0.025038806721568108,
0.028834352269768715,
0.008129462599754333,
0.0689375102519989,
-0.000727143429685384,
-0.0070396061055362225,
-0.07695275545120239,
-0.006124082952737808,
-0.04576053097844124,
-0.011152218095958233,
-0.019485346972942352,
0.09232164919376373,
-0.00945914164185524,
0.06477974355220795,
-0.00464084604755044,
0.01707041636109352,
0.07057929784059525,
0.0419575534760952,
0.06855539232492447,
0.025468580424785614,
-0.04791630804538727,
-0.025878475978970528,
0.028024310246109962,
-0.000923727173358202,
-0.07437430322170258,
0.07423824071884155,
-0.053865522146224976,
0.016568215563893318,
-0.02614372782409191,
0.05729737877845764,
-0.008956553414463997,
0.021816598251461983,
-0.04738137125968933,
0.07746128737926483,
0.04995870962738991,
-0.012032214552164078,
-0.014036583714187145,
0.007724254857748747,
-0.0688769593834877,
0.011562242172658443,
0.012766948901116848,
-0.042590752243995667,
0.019753625616431236,
0.010075431317090988,
-0.026411330327391624,
0.047374188899993896,
0.049710821360349655,
-0.03215907886624336,
0.07086241245269775,
0.01178594958037138,
-0.011751418001949787,
-0.10300008207559586,
0.07491286098957062,
-0.008991723880171776,
-0.08048240095376968,
-0.0995195284485817,
0.008495613932609558,
0.057764697819948196,
-0.03071056492626667,
0.05677402764558792,
-0.040577229112386703,
0.07169657200574875,
-0.013771271333098412,
0.03392905741930008,
-0.04796967655420303,
0.06542499363422394,
6.483928685357444e-32,
-0.01995350793004036,
0.03948189318180084,
-0.06662756949663162,
0.013018524274230003,
0.010272148065268993,
0.013908185996115208,
-0.028529474511742592,
0.09098910540342331,
0.029819002375006676,
-0.021224871277809143,
0.0652141124010086,
0.051959868520498276,
-0.05690596625208855,
0.000024641232812427916,
-0.04390662908554077,
-0.026995884254574776,
-0.0772026926279068,
0.03808628022670746,
-0.009256979450583458,
-0.0016164523549377918,
0.013259266503155231,
-0.01757672056555748,
-0.03566031530499458,
0.03102714940905571,
0.01799234375357628,
0.07733505219221115,
-0.08267918974161148,
0.04719981178641319,
0.052395064383745193,
0.05667295679450035,
0.14446577429771423,
-0.00044115004129707813,
0.006959172431379557,
-0.05465293303132057,
0.11198548972606659,
-0.008894185535609722,
0.013252528384327888,
-0.03107094205915928,
-0.02961980737745762,
-0.01264116633683443,
-0.023184925317764282,
0.016048138961195946,
-0.06516857445240021,
-0.000653197115752846,
0.028190869837999344,
-0.05213317647576332,
0.011896379292011261,
-0.031732138246297836,
0.012339895591139793,
-0.05376385152339935,
-0.060616713017225266,
0.04419010132551193,
-0.008363151922821999,
0.04406420513987541,
-0.0007092655287124217,
-0.0385279580950737,
-0.03702767938375473,
-0.031438253819942474,
0.04464888200163841,
-0.028090082108974457,
0.04317638650536537,
0.11258652061223984,
-0.09515447169542313,
-0.047746043652296066
] |
39
Equations and inequalities
3.1 Linear simultaneous equations
Linear simultaneous equations in two unknowns have one set of values that will make a pair of
equations true at the same time.
The solution to this pair of simultaneous equations is x = 5, y = 2
x + 3y = 11 (1)
4x
β 5y = 10 (2)
β Linear simultaneous equations can be solved using elimination or substit
ution.5 + 3(2) = 5 + 6 = 11 β
4(5) β 5(2) = 20 β 10 = 10 β
Example 1
Solve the simultaneous equations:
a 2x
+ 3y = 8 b 4x
β 5y = 4
3x
β y = 23 6x
+ 2y = 25
a 2x + 3 y = 8 (1)
3x
β y = 23 (2)
9x
β 3y = 69 (3)
11x
= 77
x = 7
14 + 3 y = 8
3y = 8 β 14
y = β2
The solution is x = 7, y = β 2.
b 4x β
5y = 4 (1)
6x
+ 2y = 25 (2)
12x
β 15 y = 12 (3)
12x
+ 4y = 50 (4)
β19y
= β38
y = 2
4x β 10 = 4
4x = 14
x = 3 1 __ 2
The s
olution is x = 3 1 __ 2 , y = 2.Remember to check your solution by substituting
into equation (2). 3(7) β (β2) = 21 + 2 = 23 β
Note that you could also multiply equation (1)
by 3 and equation (2) by 2 to get 6x in both equations. You could then subtract to eliminate x.
Multiply equation (1) by 3 and multiply equation (2) by 2 to get 12x in each equation.
Subtract, since the 12x terms have the same sign (both positive).
Substitute y = 2 into equation (1) to find x.First look for a way to eliminate x or y.
Multiply equation (2) by 3 to get 3y in each equation.
Number this new equation (3).
Then add equations (1) and (3), since the 3y terms
have different signs and y will be eliminated.
Substitute x = 7 into equation (1) to find y.
|
[
0.0145180094987154,
0.06115281581878662,
0.05212811753153801,
-0.035404931753873825,
-0.03482174128293991,
0.00710943341255188,
-0.05383233726024628,
-0.08176073431968689,
-0.0741613581776619,
0.038494814187288284,
-0.02935074456036091,
-0.00527762807905674,
0.039088089019060135,
0.05890903249382973,
0.008852377533912659,
0.035387665033340454,
-0.013550504110753536,
0.005598270334303379,
-0.06250534951686859,
0.0020965677686035633,
0.006531578954309225,
-0.12602970004081726,
-0.09040212631225586,
0.04423516243696213,
0.01648322306573391,
-0.058576587587594986,
-0.061368782073259354,
-0.008318844251334667,
-0.06477482616901398,
0.026526203379034996,
0.06583034992218018,
-0.05579995736479759,
0.025548480451107025,
-0.044747840613126755,
-0.0001286696788156405,
-0.04826020449399948,
0.048411305993795395,
0.0484088771045208,
0.02525738626718521,
-0.062202878296375275,
-0.02390921302139759,
-0.05616513267159462,
-0.012602303177118301,
-0.06421822309494019,
-0.018560195341706276,
-0.03239641711115837,
0.009570722468197346,
-0.006049519404768944,
0.05117388814687729,
-0.017539191991090775,
0.02112211287021637,
-0.06024935841560364,
0.033676546066999435,
0.08451010286808014,
-0.03666997328400612,
-0.07119568437337875,
-0.005797852762043476,
-0.00009777216473594308,
-0.10266444832086563,
0.04344530031085014,
-0.006349345203489065,
-0.044253479689359665,
-0.02654375694692135,
0.05007223039865494,
0.04195704683661461,
0.11651242524385452,
0.049644503742456436,
0.026277612894773483,
-0.025709567591547966,
0.03190133720636368,
-0.07004059106111526,
-0.026228616014122963,
-0.02066083624958992,
-0.03987975791096687,
0.016450483351945877,
0.026380296796560287,
-0.025387048721313477,
-0.11292482912540436,
0.015546808019280434,
-0.02355842851102352,
0.043300800025463104,
0.02172059193253517,
0.00671918410807848,
0.009624531492590904,
-0.0005128926131874323,
-0.10274965316057205,
0.012678940780460835,
0.08015061169862747,
-0.014106852002441883,
0.024514546617865562,
0.06833640486001968,
0.02242244966328144,
0.014940324239432812,
-0.022360578179359436,
0.05517754331231117,
-0.10188326239585876,
0.06248529255390167,
0.013324756175279617,
-0.0510883629322052,
0.02224380150437355,
0.0046271090395748615,
-0.09151222556829453,
0.0029073869809508324,
-0.019167805090546608,
0.023788385093212128,
0.04745259881019592,
0.007564414292573929,
-0.02595919370651245,
-0.020083818584680557,
0.0017819722415879369,
0.03664391487836838,
-0.07369987666606903,
0.07962384074926376,
0.0354311503469944,
0.013547108508646488,
-0.001841653836891055,
0.02604285441339016,
0.0055489675141870975,
0.059115804731845856,
-0.11931575834751129,
-0.14205144345760345,
-0.04656251147389412,
0.06285324692726135,
-0.03813495486974716,
-0.0312236025929451,
-0.00981155689805746,
-0.013869576156139374,
0.01209226343780756,
-0.027471430599689484,
-0.11601727455854416,
-0.06526075303554535,
0.008314737118780613,
-0.020472383126616478,
-0.06399298459291458,
-0.0028473332058638334,
-0.021923424676060677,
-0.030164144933223724,
0.04235353693366051,
-0.05330109968781471,
-0.06792601197957993,
-0.07020654529333115,
0.04024123400449753,
0.11338546127080917,
0.041834376752376556,
0.022173665463924408,
0.05733855068683624,
-0.007698678877204657,
0.015756504610180855,
-0.011073250323534012,
-0.004194370470941067,
-0.0003103814087808132,
0.02352549135684967,
0.05272486060857773,
0.057281989604234695,
-0.050883736461400986,
-0.08282975107431412,
-0.06272122263908386,
0.0006602101493626833,
-0.02817304991185665,
0.03639892116189003,
-0.006922553293406963,
0.03647686913609505,
0.0004862423811573535,
0.05085359513759613,
-0.05461488664150238,
0.045184895396232605,
0.005948306992650032,
0.04411815106868744,
-0.038743309676647186,
-0.028793850913643837,
0.05216187238693237,
0.04721453785896301,
-0.028494292870163918,
0.06843367964029312,
0.006613751873373985,
-0.015197264961898327,
0.0095157315954566,
-0.0367807075381279,
-0.07416373491287231,
0.06689654290676117,
0.016750751063227654,
-0.0801616981625557,
0.0205684881657362,
0.010773672722280025,
-0.03337681666016579,
-0.058218177407979965,
-0.012843661941587925,
-0.04941459372639656,
-0.0735451802611351,
-0.029010925441980362,
0.015414495021104813,
-0.10905445367097855,
0.0500057227909565,
0.0906774029135704,
-0.05707360804080963,
-0.0635175108909607,
0.026678774505853653,
0.005851713474839926,
-0.05958639830350876,
-0.03673722967505455,
0.03656427189707756,
0.018088681623339653,
0.08191871643066406,
-0.01037000771611929,
0.02861444652080536,
0.05740353465080261,
-0.015541153028607368,
-0.014908679760992527,
0.05239025130867958,
0.04094972461462021,
0.004864856135100126,
0.0710815042257309,
-0.04994133114814758,
0.04281796142458916,
0.043469805270433426,
-0.040263812988996506,
0.06047481670975685,
-0.02852100320160389,
-0.039200492203235626,
0.004565645940601826,
-0.03956349939107895,
0.004884502850472927,
0.1386934518814087,
-6.669725559206889e-33,
-0.007291045505553484,
0.04832275211811066,
-0.06959684938192368,
-0.07144929468631744,
-0.03019236959517002,
-0.03229277953505516,
0.06642874330282211,
-0.13901567459106445,
0.04589088261127472,
-0.008413190953433514,
0.030918333679437637,
-0.02790428139269352,
-0.045162636786699295,
0.05592907965183258,
-0.01921921782195568,
-0.0460292249917984,
-0.020541615784168243,
-0.028370007872581482,
0.0001733655808493495,
-0.0116959810256958,
0.0453011691570282,
-0.00461222417652607,
-0.05932998284697533,
-0.0274361539632082,
-0.03898179903626442,
0.03796220198273659,
0.07474871724843979,
-0.03432045504450798,
-0.03194054216146469,
0.028484225273132324,
0.028881845995783806,
-0.002862366382032633,
0.05190238729119301,
0.04231400042772293,
-0.009014951065182686,
-0.09682832658290863,
-0.03156809136271477,
-0.05596020445227623,
0.06183505058288574,
-0.028554333373904228,
0.043654486536979675,
0.06201006472110748,
0.05411700904369354,
0.022759292274713516,
0.02095915749669075,
0.07046520709991455,
0.022481005638837814,
0.05155427008867264,
-0.03715915232896805,
-0.002000780077651143,
0.020056532695889473,
0.0024345710407942533,
-0.0741305872797966,
-0.0538349486887455,
0.09317762404680252,
-0.08222053200006485,
0.0816975086927414,
-0.0210095327347517,
0.03531395271420479,
0.07794269919395447,
-0.020007392391562462,
-0.07564650475978851,
0.15945588052272797,
0.05075902119278908,
-0.043928224593400955,
0.037776537239551544,
-0.006968278903514147,
-0.007710861507803202,
0.044780928641557693,
0.016513122245669365,
0.0003910076047759503,
0.016916917636990547,
-0.025595353916287422,
-0.04333795979619026,
0.0036961629521101713,
0.037962134927511215,
-0.08946733176708221,
0.12481161206960678,
0.01962524838745594,
-0.0024197471793740988,
-0.04849373921751976,
0.03860480338335037,
-0.031403783708810806,
-0.013953096233308315,
-0.1436362862586975,
0.008333220146596432,
0.019998736679553986,
0.001390045159496367,
-0.03563893586397171,
0.0021251803264021873,
0.05026655271649361,
0.040782708674669266,
0.022513285279273987,
-0.008879337459802628,
0.11679603159427643,
6.258527055840095e-32,
-0.01864243857562542,
0.04210750386118889,
-0.062137894332408905,
0.0386740118265152,
0.02105214074254036,
0.02237248606979847,
0.02880193293094635,
0.063942089676857,
0.01710628531873226,
-0.014875806868076324,
0.07177221775054932,
0.0004287700285203755,
-0.08254238963127136,
-0.013817214407026768,
-0.04247350990772247,
-0.011671293526887894,
-0.052557386457920074,
0.048810042440891266,
-0.01764746755361557,
0.04098159074783325,
0.047371260821819305,
-0.031171387061476707,
-0.08802691847085953,
0.04589461907744408,
-0.021406788378953934,
0.06906754523515701,
-0.07721710950136185,
0.059806328266859055,
-0.005400709807872772,
0.00021267578995320946,
0.08407630026340485,
-0.06663475930690765,
-0.052631549537181854,
-0.07332316040992737,
0.08118724077939987,
0.076569564640522,
-0.01146684866398573,
0.03892894461750984,
-0.02589593641459942,
-0.03288984298706055,
0.006681795231997967,
-0.019532985985279083,
-0.06447356194257736,
0.005093560088425875,
0.007977507077157497,
-0.006019953638315201,
0.06512799113988876,
-0.12922513484954834,
0.03822340443730354,
-0.0933656319975853,
-0.08379283547401428,
0.1743011474609375,
0.007737389765679836,
0.08803670108318329,
-0.06181586906313896,
-0.015450392849743366,
-0.038554299622774124,
0.001915999804623425,
0.03124408982694149,
-0.040828242897987366,
0.04950689896941185,
0.0480206198990345,
0.016303574666380882,
-0.04410431161522865
] |
40
Chapter 3
Example 2
1 Solve these simultaneous equations by elimination:
a 2x β
y = 6 b 7x +
3y = 16 c 5x
+ 2y = 6
4x
+ 3y = 22 2x +
9y = 29 3x β
10y = 26
d 2x β
y = 12 e 3x β
2y = β6 f 3x
+ 8y = 33
6x
+ 2y = 21 6x +
3y = 2 6x =
3 + 5y
2 Solve these simultaneous equa
tions by substitution:
a x +
3y = 11 b 4x β
3y = 40 c 3x
β y = 7 d 2y
= 2x β 3
4x β
7y = 6 2x +
y = 5 10x +
3y = β2 3y
= x β 1
3 Solve these simultaneous equa
tions:
a 3x β
2y + 5 = 0 b x β 2
y ______ 3 = 4 c 3y = 5(x β 2)
5(x
+ y) = 6(x + 1) 2x +
3y + 4 = 0 3(x β
1) + y + 4 = 0
4 3x
+ ky = 8
x β
2ky = 5
are simultaneous equa
tions where k is a constant.
a Show that
x = 3. (3 marks)
b Given tha
t y = 1 _ 2 determine the value of k. (1 mark)
5 2x
β py = 5
4x
+ 5y + q = 0
are simultaneous equa
tions where p and q are constants.
The solution to this pair of simultaneous equa
tions is x = q, y = β1.
Find the value of
p and the value of q. (5 marks) First rearrange
bo
th equations into
the same form
e.g. ax + by = c.Hint
E/P
k is a constant, so it has the same value in both equations.Problem-solving
E/PExercise 3ASolve the simultaneous equations:
2x β y = 1
4x + 2y = β30
2x β y = 1 (1)
4x
+ 2y = β30 (2)
y
= 2x β 1
4x + 2(2 x β 1) = β 30
4x + 4 x β 2 = β 30
8x = β28
x = β3 1 __ 2
y =
2(β3 1 __ 2 ) β 1 = β8
The solution is x = β 3 1 __ 2 , y = β8.Rearrange an equation, in this case equation (1),
to get either x = β¦ or y = β¦ (here y = β¦).
Solve for x.
Substitute x = β3 1 _ 2 into equation (1) to find the
value o
f y.Substitute this into the other equation (here into
equation (2) in place of y).
Remember to check your solution in equation (2). 4(β3.5) + 2(β8) = β14 β 16 = β30 β
|
[
-0.03267097845673561,
0.0317545086145401,
-0.0309017114341259,
-0.0041029485873878,
-0.012019617483019829,
0.07181601971387863,
-0.04162114858627319,
-0.03919157013297081,
-0.034522343426942825,
0.05312203988432884,
0.00486091710627079,
0.04899703711271286,
0.04045356065034866,
-0.05981069058179855,
0.03181290253996849,
-0.021038932725787163,
-0.06552693992853165,
0.035113852471113205,
-0.04203997924923897,
-0.0016769716748967767,
0.05461692437529564,
-0.07451138645410538,
-0.11154539883136749,
0.013697960413992405,
0.03641920164227486,
-0.053269557654857635,
-0.04085582122206688,
0.005185470450669527,
-0.009410832077264786,
-0.054522331804037094,
0.08271495252847672,
0.05380181223154068,
-0.014387622475624084,
0.011174525134265423,
0.02294716238975525,
-0.011183511465787888,
0.030474364757537842,
0.030982406809926033,
-0.04018956422805786,
-0.08808118104934692,
-0.11434007436037064,
-0.03277640417218208,
-0.06743186712265015,
0.01629367470741272,
0.030613895505666733,
-0.0056611318141222,
-0.03846409544348717,
0.038057222962379456,
0.030579645186662674,
-0.09029757231473923,
0.021627629175782204,
-0.06351597607135773,
0.0033267003018409014,
0.016083305701613426,
0.004619559273123741,
-0.051411084830760956,
-0.010355538688600063,
0.07260897755622864,
-0.04379687458276749,
-0.006400050595402718,
-0.015631703659892082,
0.012130980379879475,
-0.03195924684405327,
0.051331717520952225,
0.037013448774814606,
0.07187165319919586,
0.04789632931351662,
-0.04367316886782646,
-0.012073642574250698,
0.06186628341674805,
-0.08869694173336029,
0.011797877959907055,
-0.08448051661252975,
-0.013064814731478691,
0.07718362659215927,
0.03441588208079338,
0.036918554455041885,
-0.03333795443177223,
-0.002413826994597912,
-0.016778169199824333,
0.03353571146726608,
0.010362562723457813,
0.052551526576280594,
0.0010861470364034176,
0.05240941420197487,
-0.07847531139850616,
-0.06201328709721565,
-0.000909628055524081,
0.05197282135486603,
-0.03858482092618942,
0.04036139324307442,
-0.03531752526760101,
0.0038251280784606934,
-0.09883099049329758,
0.014428215101361275,
-0.05436917394399643,
0.03446550667285919,
-0.0034015830606222153,
0.03887690231204033,
0.028276260942220688,
0.03837069123983383,
-0.05624859035015106,
0.014053293503820896,
0.007051367778331041,
0.011858866550028324,
-0.009588039480149746,
0.015312244184315205,
-0.06927390396595001,
0.08825095742940903,
-0.06331901997327805,
-0.04052736237645149,
-0.08965875208377838,
0.01962917111814022,
-0.013721632771193981,
-0.00564109580591321,
-0.012545405887067318,
0.033083561807870865,
-0.0004805101780220866,
0.023649081587791443,
-0.11628790944814682,
-0.008313368074595928,
-0.022475866600871086,
0.0655972883105278,
-0.011376934126019478,
-0.021917633712291718,
-0.017748503014445305,
-0.08769676834344864,
0.009422305971384048,
-0.10712281614542007,
-0.09246287494897842,
-0.017616888508200645,
-0.0779402107000351,
-0.024754371494054794,
-0.013187393546104431,
0.000034692417102633044,
0.004907628521323204,
0.0009571797563694417,
0.01766766607761383,
-0.03221708908677101,
-0.07133801281452179,
0.007243169005960226,
0.061815038323402405,
0.04996605962514877,
-0.02511056512594223,
0.017556682229042053,
0.038604795932769775,
-0.019472168758511543,
0.005835340358316898,
0.03169989958405495,
0.049501772969961166,
0.02908524125814438,
0.02026989683508873,
0.028546607121825218,
0.1148819550871849,
-0.03937976807355881,
-0.008089466020464897,
-0.040386464446783066,
-0.055946893990039825,
-0.009378327056765556,
-0.0100301718339324,
-0.06937965750694275,
0.047071658074855804,
-0.08195014297962189,
0.05381222441792488,
-0.0194754209369421,
0.0005739299231208861,
-0.0014196651754900813,
0.0664936751127243,
-0.05147873982787132,
-0.020496925339102745,
0.06120489910244942,
-0.007409418933093548,
-0.01455596648156643,
0.06838793307542801,
0.03630988672375679,
0.030253976583480835,
0.05659019947052002,
-0.031754519790410995,
-0.053725842386484146,
0.07636355608701706,
-0.003222605213522911,
-0.06068958714604378,
-0.004978150594979525,
0.03987127169966698,
0.018459482118487358,
-0.05451856181025505,
-0.05614251643419266,
0.06037891283631325,
-0.056620389223098755,
0.048397380858659744,
0.0265060905367136,
-0.011838788166642189,
0.061092980206012726,
0.11681586503982544,
-0.08075397461652756,
-0.06124573573470116,
0.04196297004818916,
-0.011116690933704376,
-0.09125827997922897,
0.029229769483208656,
0.09571956843137741,
0.008674654178321362,
0.06536509096622467,
-0.06151719391345978,
0.04758361726999283,
-0.032190632075071335,
-0.04891016334295273,
-0.0388261042535305,
0.03890794888138771,
0.050583433359861374,
0.01391049288213253,
0.09151680767536163,
-0.0432412326335907,
-0.017512483522295952,
-0.004594466648995876,
-0.06654912233352661,
0.06908407062292099,
-0.1029525026679039,
-0.027362966910004616,
0.01492498628795147,
-0.018099170178174973,
-0.0391511470079422,
0.1704609990119934,
-1.2645243827752475e-32,
0.030718589201569557,
0.064812071621418,
-0.10386038571596146,
-0.07194741070270538,
-0.04380703717470169,
-0.036655597388744354,
0.044065408408641815,
-0.11053098738193512,
0.0877067893743515,
-0.047765083611011505,
-0.019147107377648354,
-0.017221039161086082,
-0.08039765059947968,
0.024541771039366722,
-0.06735701858997345,
-0.03213687241077423,
-0.005876357201486826,
-0.06239563971757889,
-0.03752165287733078,
-0.0046770088374614716,
-0.009878052398562431,
0.017760761082172394,
-0.010396338999271393,
-0.07046385109424591,
-0.037278492003679276,
0.013475928455591202,
0.06845571845769882,
-0.0553271509706974,
-0.02732148766517639,
0.020480703562498093,
-0.0005321331555023789,
-0.008142350241541862,
0.09291994571685791,
0.08131841570138931,
-0.03842318803071976,
-0.04831849783658981,
-0.03908119723200798,
0.022072795778512955,
-0.07923323661088943,
-0.04667432978749275,
0.06084785982966423,
0.01819680444896221,
-0.020545002073049545,
0.0143116544932127,
0.024792950600385666,
0.13169513642787933,
0.009881190955638885,
0.04111574590206146,
-0.01229240745306015,
-0.006517310161143541,
-0.022962968796491623,
-0.03500960394740105,
-0.03699967637658119,
-0.041650932282209396,
0.09484764188528061,
-0.056551434099674225,
0.119623102247715,
-0.06955525279045105,
0.07585378736257553,
0.007625706028193235,
-0.023669345304369926,
0.02476838044822216,
0.07550445199012756,
0.056798115372657776,
0.05947320535778999,
0.04274357482790947,
-0.02454219199717045,
-0.005270673427730799,
-0.034606147557497025,
-0.006725671701133251,
-0.029139358550310135,
0.0922180712223053,
-0.029702505096793175,
-0.08645343035459518,
0.10862506926059723,
0.04013806954026222,
-0.09786036610603333,
0.0645463764667511,
-0.03487667441368103,
-0.011244588531553745,
-0.100956492125988,
-0.022869402542710304,
0.02115548402070999,
-0.07069917023181915,
-0.07128919661045074,
-0.018644491210579872,
0.0857723206281662,
0.05516389384865761,
0.012043050490319729,
-0.05858984589576721,
0.053859129548072815,
-0.030568063259124756,
0.11490702629089355,
0.04931528866291046,
0.1003585159778595,
9.113540111000023e-32,
-0.00596983078867197,
-0.041173938661813736,
-0.0896664708852768,
0.045730218291282654,
0.016795461997389793,
-0.013356941752135754,
-0.02310298942029476,
0.061707764863967896,
-0.0441833920776844,
-0.01697079837322235,
0.01832638494670391,
0.05848870798945427,
-0.0483689084649086,
0.031630922108888626,
-0.038137566298246384,
-0.031002400442957878,
-0.040210824459791183,
0.034391600638628006,
-0.03357194364070892,
0.019411815330386162,
-0.0020967121236026287,
-0.017218003049492836,
-0.07303515076637268,
0.04930021986365318,
0.008687743917107582,
0.10737766325473785,
-0.04207773879170418,
0.010640697553753853,
0.011381508782505989,
0.029841147363185883,
0.10252171754837036,
-0.08324283361434937,
-0.06057284399867058,
-0.12027350813150406,
0.04881671816110611,
0.08874811977148056,
0.053319092839956284,
-0.009111903607845306,
0.03395700082182884,
-0.032496076077222824,
-0.004964937921613455,
0.025343583896756172,
-0.0616973377764225,
0.06711775809526443,
0.02807638980448246,
-0.06276576220989227,
-0.013980462215840816,
-0.03999779745936394,
0.017475048080086708,
-0.05610058456659317,
-0.05955260246992111,
0.045356251299381256,
-0.03034113347530365,
-0.014500215649604797,
0.019357191398739815,
-0.09122999012470245,
-0.0119178406894207,
0.014888395555317402,
0.03068239800632,
-0.014616918750107288,
0.006425768136978149,
0.04084131494164467,
-0.03903616592288017,
-0.048172954469919205
] |
41
Equations and inequalities
3.2 Quadratic simultaneous equations
You need to be able to solve simultaneous equations where one equation is linear and one is quadratic.
To solve simultaneous equations involving one linear equation and one quadratic equation, you need
to use a substitution method from the linear equation into the quadratic equation.
β Simultaneous equations with one linear and one quadratic equation can hav
e up to two pairs
of solutions. You need to make sure the solutions are paired correctly.
The solutions to this pair of simultaneous equations are x = 4, y = β3 and x = 5.5, y = β1.5.
x β y = 7 (1)
y2 + xy + 2x = 5 (2)4 β (β3) = 7 β and 5.5 β (β1.5) = 7 β
(β3)2 + (4)(β3) + 2(4) = 9 β 12 + 8 = 5 β and
(β1.5)2 + (5.5)(β 1.5) + 2(5.5) = 2.25 β 8.25 + 11 = 5 β
Example 3
Solve the simultaneous equations:
x + 2y = 3
x2 + 3xy = 10
x + 2 y = 3 (1)
x2 + 3 xy = 10 (2)
x
= 3 β 2 y
(3 β 2 y)2 + 3y(3 β 2 y) = 10
9 β 12 y + 4 y2 + 9 y β 6 y2 = 10
β2y2 β 3y β 1 = 0
2y2 + 3y + 1 = 0
(2y + 1)( y + 1) = 0
y = β 1 __ 2 or y = β 1
So
x = 4 or x = 5
Solutions are x = 4, y = β 1 __ 2
and
x = 5, y = β 1.Rearrange linear equation (1) to get x = β¦ or
y = β¦ (here x = β¦).
Substitute this into quadratic equation (2) (here in place of x ).
Solve for y using factorisation.
Find the corresponding x-values by substituting the y-values into linear equation (1), x = 3 β 2y.
There are two solution pairs for x and y.The quadratic equation can contain terms involving y
2 and xy.
(3 β 2 y)2 means (3 β 2 y)(3 β 2 y) β Section 1.2
1 Solve the simultaneous equations:
a x +
y = 11 b 2x
+ y = 1 c y =
3x
xy =
30 x2 + y2 = 1 2y2 β xy = 15
d 3a
+ b = 8 e 2u
+ v = 7 f 3x
+ 2y = 7
3a2 + b2 = 28 uv = 6 x2 + y = 8
2 Solve the simultaneous equa
tions:
a 2x
+ 2y = 7 b x +
y = 9 c 5y
β 4x = 1
x2 β 4y2 = 8 x2 β 3xy + 2y2 = 0 x2 β y2 + 5x = 41Exercise 3B
|
[
0.07885614037513733,
0.05659138038754463,
0.057167209684848785,
-0.004681245423853397,
-0.049302250146865845,
0.00618823291733861,
-0.05632118508219719,
-0.0140976682305336,
-0.08857925236225128,
0.028829822316765785,
-0.017571184784173965,
-0.04615102335810661,
0.04656057059764862,
0.05875536799430847,
0.06456975638866425,
0.006826643366366625,
-0.04925213009119034,
-0.02271907776594162,
-0.05795888602733612,
-0.017331605777144432,
0.014566246420145035,
-0.11033613979816437,
-0.11974731832742691,
-0.03580808266997337,
0.034459829330444336,
-0.0326421856880188,
-0.08563488721847534,
-0.058599650859832764,
-0.019945302978157997,
0.06690485030412674,
0.05370859429240227,
-0.0048831794410943985,
0.019181882962584496,
-0.020501969382166862,
0.04490178823471069,
-0.05273580551147461,
0.11094965040683746,
0.08098725974559784,
-0.012604677118360996,
-0.09286810457706451,
0.0007593210902996361,
-0.011261940002441406,
-0.04785063490271568,
-0.013782587833702564,
-0.020473549142479897,
-0.031794555485248566,
-0.0024559814482927322,
-0.0521608367562294,
0.08783842623233795,
0.007285137660801411,
0.023527322337031364,
-0.006290798541158438,
0.0036668137181550264,
0.06303824484348297,
-0.03166242688894272,
-0.03946222364902496,
-0.019611719995737076,
-0.014178008772432804,
-0.05513133481144905,
0.04634832218289375,
0.056277744472026825,
0.010442441329360008,
-0.009657753631472588,
0.055680640041828156,
0.035744618624448776,
0.054313331842422485,
0.06025032699108124,
0.026752708479762077,
-0.03802650794386864,
0.1253097504377365,
-0.06923732161521912,
-0.004804571159183979,
-0.03323907405138016,
-0.037200964987277985,
0.017630919814109802,
0.019483597949147224,
-0.021991340443491936,
-0.0227045938372612,
-0.03645046800374985,
-0.01977798342704773,
0.051948145031929016,
0.0682549849152565,
0.032887984067201614,
-0.02401134744286537,
-0.02825133502483368,
-0.09612227231264114,
0.03213224187493324,
0.12034284323453903,
-0.026298558339476585,
0.06951598823070526,
0.007597736082971096,
0.015756793320178986,
0.019914239645004272,
-0.030956171452999115,
0.029780887067317963,
-0.11389071494340897,
0.05395098403096199,
0.004143886733800173,
-0.009736322797834873,
-0.005816692020744085,
-0.013807401061058044,
-0.14447809755802155,
0.023479187861084938,
-0.03266626596450806,
0.023365125060081482,
0.03006548434495926,
0.04373988136649132,
-0.035787392407655716,
-0.006850617006421089,
-0.030702024698257446,
-0.023543156683444977,
-0.07074416428804398,
0.03173381835222244,
0.025150461122393608,
0.011249377392232418,
0.00026239381986670196,
0.03537415340542793,
-0.0007466479437425733,
0.050488632172346115,
-0.11691928654909134,
-0.04984521120786667,
0.0015176758170127869,
0.1122056245803833,
-0.07239988446235657,
-0.028988109901547432,
0.009472387842833996,
-0.040022023022174835,
0.012385686859488487,
-0.05725031718611717,
-0.07431506365537643,
-0.0029652088414877653,
-0.07682125270366669,
-0.053951479494571686,
-0.08975676447153091,
-0.0027330329176038504,
0.008703555911779404,
-0.043550215661525726,
0.05602674558758736,
0.004051836673170328,
-0.052447739988565445,
-0.020657097920775414,
-0.005074166692793369,
0.07733680307865143,
0.04039175808429718,
0.025607846677303314,
0.013946348801255226,
-0.0021594318095594645,
0.03391926735639572,
-0.036022596061229706,
-0.018635429441928864,
-0.04079873859882355,
0.021594172343611717,
0.0439126081764698,
0.10297287255525589,
-0.0005025556311011314,
-0.05994708836078644,
-0.03827830031514168,
0.024882882833480835,
-0.02709893323481083,
0.007929167710244656,
-0.06304977834224701,
0.040804423391819,
0.00538818072527647,
0.02238488756120205,
-0.025186022743582726,
0.03177279978990555,
-0.03396657481789589,
0.033892951905727386,
-0.002185855759307742,
-0.059635091572999954,
0.028106365352869034,
0.0065126679837703705,
0.006561689544469118,
0.07340524345636368,
-0.03202233463525772,
-0.02011093683540821,
0.045704517513513565,
-0.006319915875792503,
-0.07265423238277435,
0.06413765251636505,
-0.011753399856388569,
-0.08491060137748718,
-0.002936922013759613,
0.014867444522678852,
0.08362387120723724,
-0.047417860478162766,
0.051266442984342575,
-0.02545197866857052,
-0.06878484785556793,
-0.06411182135343552,
-0.006998714525252581,
-0.11500181257724762,
0.08032193779945374,
0.11314025521278381,
-0.06795123964548111,
-0.04690742492675781,
0.04198083654046059,
0.026559697464108467,
-0.07052003592252731,
-0.03270535171031952,
0.030381012707948685,
-0.043880660086870193,
0.07742898911237717,
-0.004939236678183079,
0.045173320919275284,
-0.0036360621452331543,
-0.0502643808722496,
0.017972053959965706,
0.022815318778157234,
0.023722346872091293,
-0.017356786876916885,
0.04997055232524872,
-0.0332082137465477,
-0.027361515909433365,
-0.04603493958711624,
-0.06651298701763153,
0.029770854860544205,
-0.019898822531104088,
-0.051643211394548416,
0.04554225131869316,
-0.08376877009868622,
0.029696399345993996,
0.0964142307639122,
-6.899265075775926e-33,
-0.027311069890856743,
0.03809824213385582,
-0.09811990708112717,
-0.040876805782318115,
-0.03532073646783829,
0.04725568741559982,
0.05804508179426193,
-0.10789445787668228,
0.04553118720650673,
0.0077656228095293045,
0.09167088568210602,
-0.03271155804395676,
-0.011215167120099068,
0.04298416152596474,
-0.03337428346276283,
-0.05310097336769104,
-0.044243473559617996,
-0.042112819850444794,
0.012218808755278587,
-0.07160775363445282,
0.05479820817708969,
0.01361236535012722,
0.011045274324715137,
-0.008977225981652737,
-0.049078963696956635,
0.04036327451467514,
0.06631842255592346,
-0.01778828538954258,
-0.033684514462947845,
0.06764334440231323,
-0.009519864805042744,
-0.042501673102378845,
0.043501075357198715,
-0.009710734710097313,
0.0049457126297056675,
-0.07256413996219635,
-0.08417593687772751,
-0.06488185375928879,
0.06309521943330765,
-0.06550033390522003,
0.020909970626235008,
0.01474351342767477,
0.0696864053606987,
0.01464514434337616,
0.03144411742687225,
0.057695917785167694,
0.01970682106912136,
0.02858278900384903,
0.010161240585148335,
-0.006811731494963169,
0.00600891187787056,
-0.003387881675735116,
-0.05930638313293457,
-0.01143110916018486,
0.07167886942625046,
-0.09546222537755966,
0.043695755302906036,
0.008045020513236523,
0.04174099490046501,
0.00295963860116899,
0.05460069701075554,
-0.06349247694015503,
0.1524471640586853,
0.04139282554388046,
-0.057572633028030396,
0.050233665853738785,
0.04833812266588211,
-0.029369577765464783,
0.0855950340628624,
0.041213806718587875,
-0.022121040150523186,
-0.012526188977062702,
-0.014493091031908989,
-0.00883173756301403,
-0.05846511945128441,
0.02872365340590477,
-0.10281446576118469,
0.10064180195331573,
0.07927582412958145,
-0.036876484751701355,
-0.08935610949993134,
0.08884432911872864,
-0.03817280754446983,
-0.011873848736286163,
-0.13715042173862457,
-0.00036865356378257275,
0.03812357410788536,
0.052835963666439056,
0.023884115740656853,
0.00005277455784380436,
-0.0013014065334573388,
0.027677439153194427,
0.032477833330631256,
-0.03810715302824974,
0.03297796845436096,
6.533586268713581e-32,
-0.0515788309276104,
0.049585290253162384,
-0.07467896491289139,
0.0280567966401577,
-0.04484289884567261,
0.011019525118172169,
-0.02269328385591507,
0.05836138129234314,
0.006233301945030689,
-0.024239901453256607,
0.04306543245911598,
0.0222253929823637,
-0.11846182495355606,
-0.004570875782519579,
-0.07635747641324997,
0.023709053173661232,
-0.02162577211856842,
0.026619771495461464,
-0.03643948957324028,
0.002120052929967642,
0.008460703305900097,
0.03875095397233963,
-0.0628901794552803,
0.04684092104434967,
0.02397746406495571,
0.07217726856470108,
-0.08610492199659348,
-0.0018822127021849155,
0.0011856741039082408,
-0.004983109887689352,
0.07733814418315887,
-0.02209615148603916,
-0.030479026958346367,
-0.06741651147603989,
0.12698668241500854,
0.017916785553097725,
-0.028562217950820923,
-0.016093960031867027,
-0.005982828326523304,
-0.01885395310819149,
0.012535101734101772,
-0.013042014092206955,
-0.03911222517490387,
0.012301156297326088,
-0.008219012059271336,
-0.03785262629389763,
0.061406459659338,
-0.14133624732494354,
0.0814872458577156,
-0.011930490843951702,
-0.007877582684159279,
0.1493261456489563,
0.02521989867091179,
0.07671355456113815,
0.0008233148255385458,
-0.049381323158741,
-0.004764402285218239,
-0.020155178382992744,
0.055188506841659546,
-0.010926290415227413,
-0.006061541847884655,
0.056166402995586395,
-0.052570875734090805,
-0.07054128497838974
] |
42
Chapter 3
3 Solve the simultaneous equa
tions, giving your answers in their simplest surd form:
a x β
y = 6 b 2x
+ 3y = 13
xy =
4 x2 + y2 = 78
4 Solve the simultaneous equa
tions:
x +
y = 3
x2 β 3y = 1 (6 marks)
5 a By eliminating
y from the equations
y = 2 β 4x
3x2 + xy + 11 = 0
show that x2 β 2x β 11 = 0. (2 marks)
b Hence, or otherwise, solv
e the simultaneous equations
y = 2 β 4x3x
2 + xy + 11 = 0
giving your answers in the form a Β± b ββ―__
3 , where a and b are integers. (5 marks)
6 One pair of solutions for the sim
ultaneous equations
y = kx β 5
4x2 β xy = 6
is (1, p) where k and p are constants.
a Find the values of
k and p.
b Find the second pair of solutions for the sim
ultaneous equations. Use b rackets when you are
substituting an expression into an equation.Watch out
E/P
E/P
P
If (1, p ) is a solution, then x = 1, y = p
satisfies both equations.Problem-solving
3.3 Simultaneous equations on graphs
You can represent the solutions of simultaneous equations graphically. As every point on a line or
curve satisfies the equation of that line or curve, the points of intersection of two lines or curves satisfy both equations simultaneously.
β
The solutions to a pair o
f simultaneous equations represent the points of intersection of
their graphs.
Example 4
a On the same axes, dr aw the graphs of:
2x + 3y = 8
3x β y = 23
b Use your gra
ph to write down the solutions to the simultaneous equations.y β x = k
x2 + y2 = 4
Given that the simultaneous equations have exactly one pair of solutions, show that
k = Β± 2 ββ―__
2 Challenge
|
[
-0.030257467180490494,
0.06690631061792374,
-0.021385326981544495,
-0.003663928946480155,
0.002427809638902545,
0.07640144973993301,
-0.05000953748822212,
-0.06251165270805359,
-0.047105174511671066,
0.01979554072022438,
0.013548335991799831,
-0.015795789659023285,
0.014371208846569061,
-0.011928093619644642,
0.004659866448491812,
-0.019681574776768684,
-0.06623052060604095,
0.004867959767580032,
-0.0756194144487381,
-0.047789718955755234,
0.011118139140307903,
-0.07260891795158386,
-0.042848192155361176,
-0.021981095895171165,
0.03463199734687805,
0.0029026095289736986,
0.01871352642774582,
-0.0120852617546916,
-0.0025330856442451477,
-0.04630361497402191,
0.06246745586395264,
0.025720441713929176,
0.037364620715379715,
0.006338002160191536,
0.04343411698937416,
0.033486414700746536,
0.07602721452713013,
0.0041077109053730965,
-0.0373428650200367,
-0.13746525347232819,
-0.10369569808244705,
-0.031014584004878998,
-0.07704026252031326,
0.021206336095929146,
0.010678235441446304,
-0.029591837897896767,
-0.08213696628808975,
0.0638946145772934,
0.037763964384794235,
-0.019303489476442337,
0.03175395354628563,
-0.08108044415712357,
-0.040392812341451645,
0.0076557728461921215,
0.015869762748479843,
-0.030569542199373245,
-0.012069858610630035,
0.05567822977900505,
-0.06706707924604416,
0.019856316968798637,
0.0012759367236867547,
0.006264825817197561,
-0.030136171728372574,
0.029682941734790802,
0.022262753918766975,
0.047285325825214386,
0.027909161522984505,
-0.04900800809264183,
-0.01094694435596466,
0.06344098597764969,
-0.012218482792377472,
0.048101261258125305,
-0.08482590317726135,
-0.025739187374711037,
0.07175230234861374,
0.024492990225553513,
-0.042959023267030716,
-0.037563174962997437,
-0.0024897565599530935,
0.0033739949576556683,
-0.05692591518163681,
-0.04897579550743103,
0.07703618705272675,
0.06770383566617966,
0.052406419068574905,
-0.014704429544508457,
-0.02515191026031971,
-0.009347097016870975,
0.0324990339577198,
-0.0164207573980093,
0.04748851805925369,
-0.027833377942442894,
-0.0184440016746521,
-0.08399638533592224,
0.007449572905898094,
-0.08366513252258301,
0.02631857432425022,
-0.022977501153945923,
0.049013059586286545,
0.11160010099411011,
0.037632837891578674,
-0.028795188292860985,
0.014416595920920372,
0.0011808226117864251,
-0.036538321524858475,
-0.0007686868775635958,
0.038318194448947906,
-0.06813780963420868,
0.08159442991018295,
-0.06426740437746048,
-0.04447891563177109,
-0.11788489669561386,
0.06362730264663696,
-0.00012266983685549349,
0.01410478726029396,
0.004598954692482948,
0.03388373553752899,
-0.01302777137607336,
0.007793736644089222,
-0.1545029729604721,
-0.00562002370133996,
-0.02646828070282936,
0.10506892204284668,
-0.011363314464688301,
-0.027467798441648483,
-0.052004385739564896,
-0.050131626427173615,
-0.02065601386129856,
-0.0512242391705513,
-0.05607250705361366,
0.011026224121451378,
-0.06132955849170685,
-0.09393038600683212,
-0.04148603603243828,
-0.015348488464951515,
0.06413030624389648,
-0.009658103808760643,
0.06055288389325142,
0.002245992887765169,
-0.08540178090333939,
-0.00884126964956522,
0.060491666197776794,
0.022535277530550957,
-0.04581421613693237,
0.0012185333762317896,
0.01066227350383997,
-0.010822591371834278,
0.029503874480724335,
-0.00457953754812479,
0.031006567180156708,
0.014416069723665714,
-0.025861645117402077,
0.023349594324827194,
0.08497545123100281,
-0.03426796570420265,
-0.05934185907244682,
-0.04736551642417908,
-0.06353653967380524,
-0.01943952590227127,
-0.009836984798312187,
-0.02417881414294243,
0.03444117307662964,
-0.031289976090192795,
0.10646317154169083,
0.06195548176765442,
0.03841327875852585,
-0.0017648482462391257,
0.04062923416495323,
-0.04364805296063423,
0.0002890364557970315,
0.03742494434118271,
0.0224235188215971,
-0.00878384243696928,
0.11265326291322708,
0.01998770236968994,
-0.015670452266931534,
0.11748524755239487,
-0.03535139560699463,
-0.028691044077277184,
0.05300132930278778,
0.007513800635933876,
-0.09750872105360031,
0.008593122474849224,
-0.00036358271609060466,
0.018791163340210915,
-0.02610965631902218,
-0.04375504329800606,
0.04400219768285751,
-0.0832885131239891,
0.035130277276039124,
0.01741233468055725,
-0.040388450026512146,
0.005811304319649935,
0.09837985783815384,
-0.05285875126719475,
-0.03918321058154106,
0.018024450168013573,
-0.029382238164544106,
-0.11536365747451782,
0.03793942183256149,
0.08712729811668396,
0.008024899289011955,
0.07171221822500229,
-0.035155899822711945,
0.10464364290237427,
-0.04862397909164429,
-0.0742819756269455,
-0.006007663905620575,
0.027975672855973244,
0.048318784683942795,
0.02122998796403408,
0.04041239246726036,
-0.04573523625731468,
0.01813872717320919,
0.0022374752443283796,
-0.08640247583389282,
0.0020272270776331425,
-0.10041414201259613,
-0.01966911181807518,
-0.030689634382724762,
0.017220284789800644,
-0.046814825385808945,
0.1347687542438507,
-1.119318871696032e-32,
0.029829220846295357,
0.03557556867599487,
-0.13713502883911133,
-0.0503922700881958,
-0.03763125464320183,
-0.025526411831378937,
0.08243688941001892,
-0.07948300987482071,
0.0694209411740303,
-0.009690893813967705,
0.005332105793058872,
-0.010323400609195232,
-0.014294382184743881,
0.006194583605974913,
0.014580841176211834,
-0.027489956468343735,
0.0005095392116345465,
-0.08460430800914764,
0.039050839841365814,
-0.01901286467909813,
-0.020923033356666565,
-0.008971228264272213,
-0.006846135016530752,
-0.04931070655584335,
-0.017186004668474197,
0.008931072428822517,
0.10562966763973236,
-0.08918504416942596,
-0.03832247108221054,
-0.002590552903711796,
-0.022875718772411346,
-0.048196855932474136,
0.09361130744218826,
0.040295906364917755,
-0.0495256669819355,
-0.01884479820728302,
-0.02813158743083477,
0.004744020756334066,
-0.021554308012127876,
-0.04207026585936546,
0.10406932234764099,
0.02636674977838993,
0.0022979688365012407,
0.005699636414647102,
0.01585538685321808,
0.11010192334651947,
0.037569865584373474,
0.03777459263801575,
-0.03695311397314072,
-0.012144411914050579,
-0.04657077416777611,
-0.0686921551823616,
-0.03500688076019287,
0.008717350661754608,
0.09012103080749512,
-0.09589873254299164,
0.07442554086446762,
-0.0908048078417778,
0.07811840623617172,
0.027718808501958847,
0.03331940621137619,
0.018024981021881104,
0.0723806843161583,
0.07251822203397751,
0.08987950533628464,
0.02094224840402603,
-0.01448532473295927,
-0.03172977641224861,
0.04082993045449257,
0.008420011959969997,
-0.034509073942899704,
0.06400419026613235,
-0.05971486493945122,
-0.1081719845533371,
0.05698017030954361,
0.034335359930992126,
-0.07564667612314224,
0.08873516321182251,
-0.009156512096524239,
-0.06697532534599304,
-0.11907578259706497,
0.006106184795498848,
-0.014945929870009422,
-0.03577883541584015,
-0.06459931284189224,
0.0415368378162384,
0.10804901272058487,
0.026835493743419647,
0.05163033306598663,
-0.04615654796361923,
0.0646052360534668,
0.02270897477865219,
0.01952369697391987,
-0.022010790184140205,
0.09581802040338516,
9.53149829463122e-32,
-0.02990291640162468,
-0.01633342355489731,
-0.061360690742731094,
-0.004711095243692398,
0.05470295250415802,
0.04060721769928932,
-0.015161584131419659,
0.04872851073741913,
-0.03202174976468086,
0.00380685948766768,
0.01901770941913128,
0.004239153582602739,
-0.09810809791088104,
-0.002068605972453952,
-0.019759373739361763,
-0.04743720591068268,
-0.07498975098133087,
0.03333960473537445,
-0.007595133502036333,
-0.014621390029788017,
0.030270816758275032,
0.009655473753809929,
-0.020164847373962402,
0.10876704007387161,
0.0018877635011449456,
0.06570678949356079,
-0.0727575495839119,
-0.022508760914206505,
-0.01927832141518593,
-0.04739731177687645,
0.11579548567533493,
-0.08144824206829071,
-0.055780671536922455,
-0.08279140293598175,
-0.00814548134803772,
0.04970204457640648,
0.03414192423224449,
-0.006417971104383469,
0.004742415156215429,
0.00678605679422617,
-0.03406820446252823,
0.0164172425866127,
-0.04864680394530296,
0.10828877985477448,
0.05448126420378685,
-0.03111041896045208,
-0.01119802612811327,
-0.04488302394747734,
0.03530513867735863,
-0.08133787661790848,
-0.06497304141521454,
0.008182680234313011,
-0.021630752831697464,
0.01921352930366993,
-0.003008358646184206,
-0.07106002420186996,
-0.06483352184295654,
0.05372321233153343,
0.03617971017956734,
-0.001292782137170434,
-0.019238868728280067,
0.06274963915348053,
-0.05604943633079529,
-0.053204379975795746
] |
43
Equations and inequalities
a
β2 β/four.ss01 /four.ss01 8 2 6y
x123/four.ss01
β1
β2
β3
β/four.ss012x + 3 y = 8
3x β y = 23O
b The solution is (7, β2) or x = 7, y = β2.The point of intersection is the solution to the
simultaneous equations
2x + 3y = 8
3x β y = 23
This solution matches the algebraic solution to
the simultaneous equations.
Example 5
a On the same axes, dr aw the graphs of:
2x + y = 3
y = x2 β 3x + 1
b Use your gra
ph to write down the solutions to the simultaneous equations.
a y
x
β1β2β11 234 5 β31
O23456
(β1, 5)
(2, β1)
β2β3y = x2 β 3x + 1
2x + y = 3
b The solutions are ( β1, 5 ) or x = β 1,
y = 5 and (2, β 1) or x = 2, y = β 1.There are two solutions. Each solution will have
an x-value and a y-value.
Check your solutions by substituting into both
equations.
2(β1) + (5) = β2 + 5 = 3 β and
5 = (β1)2 β 3(β1) + 1 = 1 + 3 + 1 = 5 β
2(2) + (β1) = 4 β 1 = 3 β and β1 = (2)
2 β 3(2) + 1 = 4 β 6 + 1 = β1 β
The graph of a linear equation and the graph of a quadratic equation can either:
β’ intersect twice
β’ intersect once
β’ not intersect
Aft
er substituting, you can use the discriminant of the resulting quadratic equation to determine the
number of points of intersection. Find the point of intersection
gr
aphically using GeoGebra.Online
Plot the curve and the line using
Ge
oGebra to find the two points of intersection.Online
|
[
0.037115830928087234,
0.06332310289144516,
0.001950031379237771,
0.005138216074556112,
-0.017469869926571846,
-0.01303760427981615,
-0.05614497885107994,
-0.0004703994782175869,
-0.13892047107219696,
0.0022354547400027514,
0.005433906335383654,
-0.06087993085384369,
0.03243115171790123,
0.05226317420601845,
-0.029501356184482574,
0.0161418654024601,
-0.02660663053393364,
-0.021562516689300537,
-0.03929611295461655,
-0.05111705884337425,
-0.0416310653090477,
-0.07292763143777847,
-0.0718926265835762,
-0.06158827245235443,
0.027280785143375397,
-0.09951130300760269,
-0.027733823284506798,
0.02007308043539524,
0.0024990218225866556,
0.004511643201112747,
0.09124177694320679,
-0.02064722776412964,
0.05904413014650345,
-0.0052481102757155895,
0.058134693652391434,
-0.032850511372089386,
0.10485396534204483,
0.03673214837908745,
0.04575369134545326,
-0.06502148509025574,
-0.0942709743976593,
-0.030053460970520973,
0.0068640438839793205,
0.04031600430607796,
-0.0576457642018795,
-0.0050440398044884205,
-0.025035379454493523,
0.016524193808436394,
0.02503563091158867,
-0.06087995320558548,
0.019653547555208206,
-0.03609592467546463,
0.003133210586383939,
0.06288043409585953,
0.02107327990233898,
-0.011286113411188126,
-0.009015283547341824,
-0.02661355957388878,
-0.07815910875797272,
0.04531065747141838,
0.06110123544931412,
-0.006557467393577099,
-0.04077572003006935,
0.02453329600393772,
-0.0043136952444911,
0.08786312490701675,
0.04901275038719177,
0.027532681822776794,
-0.03182489424943924,
0.09210969507694244,
-0.09781823307275772,
0.07413063198328018,
-0.10573460906744003,
-0.08947362005710602,
0.002950540278106928,
0.03977479785680771,
-0.03806508705019951,
-0.07197370380163193,
-0.03437821567058563,
-0.04747416079044342,
-0.02619783766567707,
-0.007355778943747282,
0.007721311878412962,
0.026727648451924324,
-0.015136679634451866,
-0.002718987874686718,
0.008996771648526192,
-0.0030050622299313545,
0.015080326236784458,
0.015297789126634598,
0.004388959612697363,
0.040325965732336044,
-0.0466650053858757,
-0.07893247902393341,
0.05370026081800461,
-0.14279372990131378,
0.0401739627122879,
-0.028986159712076187,
0.0016704773297533393,
0.09876807034015656,
0.009908360429108143,
-0.021092489361763,
0.026416007429361343,
-0.0027740197256207466,
0.049834996461868286,
0.059601299464702606,
0.028067123144865036,
-0.040238846093416214,
0.027413461357355118,
-0.02508968487381935,
-0.0442110076546669,
-0.09587180614471436,
0.005925741046667099,
0.044746775180101395,
-0.000694597139954567,
0.011486000381410122,
0.08408433198928833,
-0.025528065860271454,
0.0173252671957016,
-0.06059449538588524,
-0.033364150673151016,
-0.012846432626247406,
0.08818577229976654,
-0.05389238893985748,
-0.03474396839737892,
0.024406248703598976,
-0.06342349201440811,
0.02791607938706875,
0.029117686673998833,
-0.07216296344995499,
-0.004217939916998148,
-0.05871737748384476,
-0.0062919436022639275,
-0.03683170676231384,
-0.03830704092979431,
-0.03337675705552101,
-0.056408174335956573,
0.06647118180990219,
0.01913507841527462,
-0.05091678723692894,
-0.0046080308966338634,
0.05416615307331085,
0.061099547892808914,
0.004667969420552254,
0.026441533118486404,
0.030688337981700897,
-0.11710447818040848,
0.05626436695456505,
-0.05666052922606468,
0.005601630080491304,
0.009077566675841808,
-0.04550018534064293,
0.026950955390930176,
0.1237383559346199,
-0.050130825489759445,
-0.03184672072529793,
0.006344825029373169,
-0.018043965101242065,
-0.026482341811060905,
0.029762418940663338,
-0.05066356807947159,
0.05203042924404144,
-0.0019198567606508732,
0.05471689999103546,
0.06411837041378021,
0.005301678087562323,
-0.017158187925815582,
0.040619730949401855,
0.004518560599535704,
0.01995290443301201,
0.015451956540346146,
0.06483259797096252,
-0.013433714397251606,
0.07098958641290665,
-0.044225793331861496,
0.0005771871074102819,
0.09347432851791382,
-0.018390655517578125,
-0.07788147032260895,
0.09710042178630829,
0.011304507032036781,
-0.09047242999076843,
-0.007937529124319553,
-0.01025001797825098,
-0.03501478582620621,
0.0013259036932140589,
0.0182577483355999,
-0.013418447226285934,
-0.06978146731853485,
0.041774675250053406,
-0.03828975558280945,
-0.14474919438362122,
0.015902450308203697,
0.09990396350622177,
-0.1118309423327446,
-0.009448208846151829,
0.01550876535475254,
0.003070442471653223,
-0.0736331194639206,
0.020995480939745903,
0.05351957678794861,
0.013168809935450554,
0.00648913998156786,
-0.04311021789908409,
0.024903731420636177,
0.019206702709197998,
-0.05274020880460739,
0.05243055149912834,
0.010957368649542332,
-0.010075705125927925,
-0.009461614303290844,
0.04003866761922836,
-0.057225003838539124,
-0.04769653454422951,
0.02810038812458515,
-0.07247260957956314,
0.04484901577234268,
-0.08779288083314896,
-0.1177164614200592,
0.0075523206032812595,
-0.036604009568691254,
0.005705088376998901,
0.08608844131231308,
-8.859346069790426e-33,
-0.03365713730454445,
-0.011369304731488228,
-0.05850227177143097,
-0.06810428947210312,
-0.027606559917330742,
-0.0032158216927200556,
0.07958121597766876,
-0.058457084000110626,
0.042836565524339676,
0.0015699242940172553,
0.13579443097114563,
-0.016309797763824463,
-0.035172369331121445,
0.04426712170243263,
-0.027796881273388863,
-0.10678227245807648,
-0.00014810136053711176,
-0.032815489917993546,
0.0025303650181740522,
-0.01844802498817444,
0.0505080372095108,
-0.010058867745101452,
0.029157595708966255,
-0.025229087099432945,
-0.019972065463662148,
0.041484370827674866,
0.04018683359026909,
-0.09510333091020584,
-0.0458657331764698,
0.052067458629608154,
0.016659505665302277,
-0.045593198388814926,
0.06343907117843628,
0.03199915215373039,
-0.00923380721360445,
-0.04333742335438728,
-0.07505423575639725,
-0.07874405384063721,
0.08774296939373016,
-0.07991417497396469,
0.060236506164073944,
0.03874499350786209,
0.11363472789525986,
0.04267598316073418,
0.0167548805475235,
0.06276245415210724,
-0.00480589410290122,
0.05700599029660225,
0.010760550387203693,
-0.0069623845629394054,
0.01684490405023098,
-0.05905939266085625,
-0.01401445735245943,
-0.011426667682826519,
0.11286281049251556,
-0.07180260866880417,
0.02700810320675373,
-0.03083532117307186,
0.04523467272520065,
0.01594337448477745,
0.07592202723026276,
-0.05604355409741402,
0.0814887061715126,
0.012781203724443913,
-0.017615262418985367,
0.014978744089603424,
-0.002880845917388797,
-0.08880151063203812,
0.054894767701625824,
0.013859109953045845,
0.003652293235063553,
-0.03231325000524521,
-0.09329777956008911,
-0.05555271357297897,
-0.036526236683130264,
-0.014923526905477047,
-0.007006427738815546,
0.11384934186935425,
0.10259050130844116,
-0.07071132212877274,
-0.1245514526963234,
0.07985278218984604,
-0.02507973276078701,
-0.026167498901486397,
-0.07527387887239456,
0.02354489080607891,
0.057592835277318954,
0.02423713728785515,
0.09225095808506012,
-0.006037009414285421,
0.05868026614189148,
0.015595358796417713,
-0.008330564014613628,
-0.011775591410696507,
0.09190169721841812,
9.352491076152125e-32,
0.00027031972422264516,
0.055163852870464325,
-0.04727214574813843,
-0.025901658460497856,
-0.025317253544926643,
0.022224370390176773,
-0.027991870418190956,
0.05950583890080452,
0.05074550583958626,
-0.00623953714966774,
0.0692889466881752,
0.034210167825222015,
-0.10838127136230469,
0.006791748106479645,
-0.02044382505118847,
0.04679187759757042,
-0.011816642247140408,
0.012881830334663391,
0.0263209231197834,
-0.045201897621154785,
-0.07243625074625015,
0.027309896424412727,
-0.018300028517842293,
0.05483793839812279,
0.05508362129330635,
0.05647331848740578,
-0.06801202893257141,
-0.016818203032016754,
-0.007864618673920631,
-0.008667710237205029,
0.08156053721904755,
-0.029012663289904594,
-0.03427855670452118,
-0.07150956243276596,
0.10291489958763123,
0.00922730378806591,
-0.09948013722896576,
-0.005945151671767235,
0.003941040486097336,
-0.022612079977989197,
-0.007578393444418907,
-0.024934586137533188,
-0.03560159355401993,
-0.023594580590724945,
-0.003101813141256571,
-0.05454380437731743,
0.02051912434399128,
-0.07605066150426865,
0.014866775833070278,
-0.03251953795552254,
-0.06988988816738129,
0.10318145900964737,
-0.032399982213974,
0.02086246758699417,
-0.006066325586289167,
-0.04382047802209854,
-0.006725102197378874,
-0.007499347906559706,
0.020713232457637787,
-0.020231159403920174,
-0.03338371589779854,
0.07080426812171936,
-0.08221644908189774,
-0.026650553569197655
] |
44
Chapter 3
β For a pair of simultaneous equations that produce a quadratic equation of the form
ax2 + bx + c = 0:
β’ b2 β 4ac > 0 β’ b2 β 4ac = 0 β’ b2 β 4ac < 0
two real solutions one real solution no real solutions
Example 6
The line with equation y = 2x + 1 meets the curve with equation kx2 + 2y + (k β 2) = 0 at exactly
one point. Given that k is a positive constant
a find the value of
k
b for this va
lue of k, find the coordinates of
the point of intersection.
a y = 2x + 1 (1)
kx2 + 2 y + ( k β 2) = 0 (2)
kx2 + 2(2 x + 1) + ( k β 2) = 0
kx2 + 4 x + 2 + k β 2 = 0
kx2 + 4 x + k = 0
42 β 4 Γ k Γ k = 0
16 β
4k2 = 0
k2 β 4 = 0
(k
β 2)( k + 2) = 0
k
= 2 or k = β 2
So k = 2
b 2x2 + 4 x + 2 = 0
x2 + 2x + 1 = 0
(x
+ 1)(x + 1) = 0
x = β1
y
= 2(β1) + 1 = β 1
Point of intersection is ( β1, β1).Substitute y = 2x + 1 into equation (2) and
simplify the quadratic equation. The resulting quadratic equation is in the form ax
2 + bx + c = 0
with a = k, b = 4 and c = k.
Factorise the quadratic to find the values of k.
The solution is k = +2 as k is a positive constant.
Substitute k = +2 into the quadratic equation kx
2 + 4x + k = 0. Simplify and factorise to find
the x-coordinate.
Check your answer by substituting into equation (2):
2x2 + 2y = 0
2(β1)2 + 2(β1) = 2 β 2 = 0 βYou are told that the line meets the curve at exactly one point, so use the discriminant of the resulting quadratic. There will be exactly one solution, so b
2 β 4ac = 0.Problem-solving
Substitute x = β1 into linear equation (1) to find the y-coordinate. Explore how the value of k affects
the l
ine and the curve using GeoGebra.Online
|
[
-0.04485006257891655,
0.09205262362957001,
-0.035769931972026825,
0.06134475767612457,
0.014121819287538528,
-0.025500737130641937,
-0.05322924256324768,
0.018425868824124336,
-0.05087628215551376,
0.05683095380663872,
0.03860218822956085,
-0.0549909882247448,
0.031596507877111435,
-0.005948805715888739,
-0.01224603783339262,
0.007403403054922819,
-0.061504825949668884,
-0.00891040451824665,
-0.07033346593379974,
0.0698840320110321,
-0.02591453678905964,
-0.07115235179662704,
-0.06063378229737282,
-0.04674253240227699,
-0.01794118620455265,
-0.14754129946231842,
0.023592844605445862,
-0.05248173326253891,
0.01609893888235092,
-0.010949681513011456,
-0.01886645331978798,
0.06475845724344254,
0.031807731837034225,
0.08594445884227753,
0.07398304343223572,
-0.0021134186536073685,
0.06022511050105095,
0.04790439084172249,
0.06759929656982422,
-0.12235043197870255,
-0.002258838852867484,
0.00486533110961318,
-0.05521582067012787,
0.0005779452039860189,
0.07540455460548401,
-0.01946215331554413,
-0.029963294044137,
-0.05383003130555153,
0.06538918614387512,
-0.04451074078679085,
-0.0021423192229121923,
-0.04930969700217247,
-0.1266181766986847,
0.03854083642363548,
0.013192063197493553,
0.050964150577783585,
-0.06383631378412247,
0.026685234159231186,
-0.008871674537658691,
0.11056248098611832,
0.07257723808288574,
-0.0013198017841205,
0.06498946994543076,
0.08063632994890213,
0.02260417863726616,
-0.015195133164525032,
0.04135485365986824,
0.021021509543061256,
-0.0023457405623048544,
0.04833446443080902,
-0.07592055201530457,
-0.015671344473958015,
-0.012079905718564987,
-0.01890672743320465,
0.03997495397925377,
-0.011535377241671085,
-0.045446526259183884,
0.004799364600330591,
0.03997408226132393,
-0.016367968171834946,
0.03621765226125717,
-0.03310887888073921,
0.09553315490484238,
-0.03372068330645561,
-0.014172466471791267,
-0.025359561666846275,
0.03156488016247749,
-0.01852743700146675,
0.08718830347061157,
0.00025504574296064675,
0.019657090306282043,
0.01299175713211298,
-0.021961161866784096,
-0.08520989120006561,
0.006914460565894842,
-0.09320630133152008,
0.03602549433708191,
-0.016253063455224037,
0.03721698746085167,
0.10904137790203094,
0.022995853796601295,
-0.03441396728157997,
-0.05298488959670067,
0.07566551864147186,
0.047535043209791183,
0.02722618356347084,
0.016947763040661812,
-0.007495904807001352,
0.05697058513760567,
0.011408462189137936,
-0.035193853080272675,
-0.03601106256246567,
-0.011067972518503666,
0.02967967838048935,
0.07353797554969788,
0.027100341394543648,
0.11217297613620758,
-0.02428516373038292,
0.014423801563680172,
-0.11815725266933441,
-0.044834017753601074,
-0.05701807886362076,
0.05588376894593239,
-0.09829495847225189,
-0.04862280189990997,
-0.03783600404858589,
0.008094132877886295,
-0.037514761090278625,
-0.03712870925664902,
-0.042359013110399246,
0.04237712547183037,
-0.032077137380838394,
-0.06051404029130936,
-0.03495466336607933,
0.020436448976397514,
0.014918370172381401,
-0.042738743126392365,
0.08576022833585739,
-0.0034564274828881025,
-0.10044784843921661,
0.018837343901395798,
0.07514132559299469,
0.012062404304742813,
0.04236486554145813,
-0.0037935010623186827,
0.025055957958102226,
-0.05018918216228485,
-0.06399860233068466,
-0.028618039563298225,
0.015237228013575077,
0.002226660493761301,
-0.0128296734765172,
0.01767374947667122,
0.07514485716819763,
0.013203647918999195,
-0.02753787487745285,
-0.046443793922662735,
-0.05784681811928749,
0.00932876393198967,
0.0507470928132534,
-0.05740824714303017,
0.04830123111605644,
-0.06131555140018463,
0.017948653548955917,
-0.02107829414308071,
0.02168142981827259,
-0.02509891428053379,
0.06470304727554321,
0.016578136011958122,
0.05089733004570007,
0.046475596725940704,
0.06248316168785095,
0.025625940412282944,
0.05162801221013069,
-0.01908484473824501,
0.023833859711885452,
0.07129321247339249,
0.051131635904312134,
0.00024519156431779265,
0.02307569980621338,
-0.028587326407432556,
-0.01919618621468544,
0.00949010904878378,
0.014664262533187866,
0.0231813695281744,
-0.021906699985265732,
0.0437747947871685,
-0.02482456900179386,
-0.032104335725307465,
0.02936258353292942,
-0.016581885516643524,
-0.06045972928404808,
-0.053484782576560974,
0.03988203406333923,
-0.12471953779459,
-0.022608233615756035,
0.020549112930893898,
-0.011876932345330715,
-0.03947748243808746,
0.002153558423742652,
0.03333596512675285,
-0.032388586550951004,
0.03477440029382706,
0.019501978531479836,
-0.02437816932797432,
-0.038735244423151016,
-0.033350009471178055,
0.020744988694787025,
-0.017383668571710587,
-0.0076095545664429665,
0.039232440292835236,
0.01093150582164526,
-0.056844186037778854,
-0.03565564751625061,
-0.07968071103096008,
-0.09858731925487518,
-0.018120622262358665,
-0.009673621505498886,
-0.09799190610647202,
0.07371113449335098,
-0.06773250550031662,
-0.005726886913180351,
0.04604485258460045,
-2.06500102799409e-33,
0.04939667508006096,
0.048831820487976074,
-0.1286212056875229,
0.005595317576080561,
-0.05889096111059189,
0.013533354736864567,
0.013305560685694218,
-0.10423388332128525,
0.11947309225797653,
-0.0009665153920650482,
0.06644462049007416,
-0.04377587512135506,
-0.04050876572728157,
0.0661737248301506,
-0.0005147829069755971,
0.021535979583859444,
-0.04014451429247856,
0.0022022868506610394,
0.04444682598114014,
-0.03309251740574837,
0.05466499924659729,
0.042213473469018936,
0.02687661722302437,
-0.027615327388048172,
-0.002752267988398671,
0.012791512534022331,
0.06875929981470108,
-0.017703082412481308,
-0.07898566126823425,
0.10616827011108398,
-0.053184255957603455,
-0.04683619737625122,
-0.002529461868107319,
0.010354421101510525,
-0.03741158917546272,
-0.06476999074220657,
-0.019256439059972763,
-0.02768234722316265,
0.04020392522215843,
-0.06265367567539215,
0.006504221353679895,
0.059145014733076096,
0.08536286652088165,
-0.009757621213793755,
0.0815700814127922,
0.08765382319688797,
0.07040269672870636,
0.047271426767110825,
0.04327838122844696,
0.020362766459584236,
0.024233780801296234,
-0.025598663836717606,
-0.061512481421232224,
0.04863746464252472,
0.09302450716495514,
-0.038439493626356125,
-0.04907214269042015,
-0.03574661165475845,
-0.00852136593312025,
-0.09690079838037491,
-0.007317688316106796,
-0.02777589112520218,
0.03503620997071266,
0.1084357500076294,
0.016024751588702202,
0.059877123683691025,
0.001860600314103067,
0.03696363419294357,
0.01659015566110611,
-0.00937917735427618,
-0.06967411935329437,
0.014342748560011387,
-0.1461082398891449,
-0.051679693162441254,
-0.043205250054597855,
-0.018350861966609955,
-0.0773911401629448,
0.06452342867851257,
0.005876521579921246,
-0.02812838740646839,
-0.12130579352378845,
0.07906074076890945,
-0.017470087856054306,
-0.06526394933462143,
-0.06167799234390259,
0.05889863893389702,
0.033459004014730453,
0.04827212914824486,
0.04979635030031204,
-0.01699073426425457,
0.0593746192753315,
0.013471484184265137,
-0.05281144380569458,
-0.02584845945239067,
0.06161479651927948,
7.94523567137454e-32,
-0.04210657626390457,
0.015999486669898033,
-0.04698539152741432,
0.0016442264895886183,
0.057188261300325394,
0.047612011432647705,
-0.031192131340503693,
-0.0370008684694767,
-0.028627954423427582,
-0.013569536618888378,
0.04879621043801308,
-0.04341275990009308,
-0.06802679598331451,
0.03936810791492462,
-0.08698535710573196,
0.05007888376712799,
-0.060895442962646484,
0.03546317666769028,
-0.02662411518394947,
-0.03665180131793022,
-0.043952926993370056,
-0.0047602299600839615,
-0.03458770737051964,
0.07848619669675827,
0.025828834623098373,
0.08615189045667648,
-0.0507529154419899,
0.03086998127400875,
-0.024203922599554062,
-0.029039885848760605,
0.10000018775463104,
-0.025468742474913597,
0.0294279083609581,
-0.03216013312339783,
0.05308090150356293,
-0.015488008968532085,
-0.012990504503250122,
-0.016620812937617302,
0.004066502675414085,
-0.0031805492471903563,
0.015334399417042732,
-0.04230032116174698,
-0.09046461433172226,
0.02851777710020542,
-0.032740212976932526,
0.03811015188694,
-0.0016774798277765512,
-0.13296891748905182,
0.010792667977511883,
0.002069046488031745,
-0.07005470246076584,
0.053415264934301376,
0.05639871209859848,
0.06834745407104492,
0.018014416098594666,
-0.11433003842830658,
-0.08438187837600708,
-0.021552661433815956,
0.03606921061873436,
-0.003338099457323551,
-0.09823143482208252,
0.052249230444431305,
-0.1030043363571167,
-0.04899047315120697
] |
45
Equations and inequalities
1 In each case:
i draw the gr
aphs for each pair of equations on the same axes
ii find the coordinates of
the point of intersection.
a y =
3x β 5 b y =
2x β 7 c y =
3x + 2
y =
3 β x y =
8 β 3x 3x
+ y + 1 = 0
2 a Use graph paper to draw accurately the graphs of 2y = 2x + 11 and y = 2x2 β 3x β 5 on the same axes.
b Use your graph to find the coordinates of the points of intersection.
c Verify your solutions b
y substitution.
3 a On the same axes sketch the curv
e with equation x2 + y = 9 and the line with equation 2x + y = 6.
b Find the coordinates of
the points of intersection.
c Verify your solutions b
y substitution.
4 a On the same axes sketch the curv
e with equation
y = (x β 2)2 and the line with equation y = 3x β 2.
b Find the coordinates of
the point of intersection.
5 Find the coordinates of
the points at which the line with equation y = x β 4 intersects the curve
with equation y2 = 2x2 β 17.
6 Find the coordinates of
the points at which the line with equation y = 3x β 1 intersects the curve
with equation y2 = xy + 15.
7 Determine the number of
points of intersection for these pairs of simultaneous equations.
a y =
6x2 + 3x β 7 b y = 4x2 β 18x + 40 c y = 3x2 β 2x + 4
y =
2x + 8 y =
10x β 9 7x
+ y + 3 = 0
8 Given the sim
ultaneous equations
2x β y = 1x
2 + 4ky + 5k = 0
where k is a non-zero constant
a show that
x2 + 8kx + k = 0. (2 marks)
Given tha
t x2 + 8kx + k = 0 has equal roots,
b find the value of
k (3 marks)
c for this va
lue of k, find the solution of the simultaneous equations. (3 marks)
9 A swimmer div
es into a pool. Her position, p m, underwater can be mode
lled
in relation to her horizontal distance, x m, from the point she entered the
water as a quadratic equation p = 1 _ 2 x2 β 3x.
The position of the bottom of the pool can be modelled by the linear
equation p = 0.3x β 6.
Determine whether this model predicts that the swimmer will touch the
bottom of the pool. (5 marks) You need to use algebra in
par
t b to find the coordinates.Hint
P
E/P
E/P p
xExercise 3C
|
[
0.02118661254644394,
0.06572654843330383,
0.06772848963737488,
-0.03258894756436348,
-0.03530095890164375,
0.030761877074837685,
-0.03345591202378273,
-0.01703377068042755,
-0.13323159515857697,
0.025881079956889153,
-0.023392438888549805,
-0.0253805760294199,
0.012420202605426311,
0.045523449778556824,
-0.06263437122106552,
0.035952165722846985,
0.00364526454359293,
-0.041283901780843735,
-0.030284706503152847,
-0.08385544270277023,
-0.05288437753915787,
-0.06413678824901581,
-0.06234213709831238,
-0.04131840541958809,
0.04264635965228081,
-0.06724537163972855,
-0.0333772711455822,
-0.00030797653016634285,
-0.03419983759522438,
0.007397535257041454,
0.08238478749990463,
-0.06582152843475342,
0.06948203593492508,
-0.007510358467698097,
0.06061968579888344,
-0.002930539660155773,
0.10633419454097748,
0.018827391788363457,
0.07028864324092865,
-0.07197974622249603,
-0.016416577622294426,
-0.02440045215189457,
-0.0031117189209908247,
0.003756236983463168,
0.010989371687173843,
0.02476753294467926,
-0.0496642105281353,
-0.011881123296916485,
0.05001744627952576,
-0.019217321649193764,
0.0070485640317201614,
-0.033206451684236526,
0.049309004098176956,
0.017267687246203423,
0.0058754864148795605,
-0.047883741557598114,
0.03586960211396217,
-0.037213053554296494,
-0.04522300511598587,
0.05694800242781639,
0.03615909442305565,
0.014084506779909134,
-0.06942394375801086,
0.0427531935274601,
-0.003402285510674119,
0.08570902049541473,
0.1297735720872879,
0.07235196232795715,
-0.06896843761205673,
0.11870313435792923,
-0.07954226434230804,
0.050902221351861954,
-0.02754082717001438,
-0.09796655178070068,
0.042407430708408356,
0.026470862329006195,
-0.020801082253456116,
-0.09213422238826752,
-0.02918851003050804,
-0.03167077898979187,
0.014572575688362122,
0.03515281900763512,
0.024788152426481247,
0.04734841734170914,
-0.03426434472203255,
-0.06680140644311905,
0.02969759702682495,
0.03291008248925209,
0.009881358593702316,
0.02398339845240116,
0.05007840692996979,
0.06702318787574768,
-0.04044783115386963,
-0.06825608760118484,
0.06288634240627289,
-0.1509983241558075,
0.06559126824140549,
-0.034122731536626816,
-0.012443151324987411,
0.08036402612924576,
-0.0022561270743608475,
-0.07410675287246704,
-0.021164150908589363,
0.037549328058958054,
0.06453033536672592,
0.02529698796570301,
0.007453197613358498,
-0.05919772759079933,
-0.023972036316990852,
-0.005548113491386175,
-0.010900604538619518,
-0.07848582416772842,
0.011420348659157753,
0.05800195410847664,
0.007432978600263596,
-0.003755116369575262,
0.04472479596734047,
-0.028935309499502182,
0.04111438989639282,
-0.05525556579232216,
-0.0686282217502594,
-0.04409243166446686,
0.04409322887659073,
-0.028040431439876556,
-0.0775168389081955,
0.0027247213292866945,
-0.03536369279026985,
0.03805582597851753,
0.015309912152588367,
-0.0728476271033287,
-0.02177586406469345,
-0.03644951432943344,
0.02300897054374218,
-0.03797227144241333,
-0.02023456245660782,
-0.06609705835580826,
-0.024370482191443443,
0.0014399162027984858,
-0.02739492431282997,
-0.02611362934112549,
0.011205053888261318,
0.06052648276090622,
0.052621982991695404,
0.05196444317698479,
0.044119272381067276,
0.008885563351213932,
-0.11512335389852524,
0.04512923210859299,
-0.03224177658557892,
-0.032198384404182434,
0.007666567340493202,
-0.01209168415516615,
0.03978521376848221,
0.10670160502195358,
-0.043877847492694855,
-0.041077058762311935,
-0.017553411424160004,
0.0017119671683758497,
0.0034572966396808624,
0.026611613109707832,
-0.02910846285521984,
0.09405014663934708,
-0.03467567637562752,
0.05362638831138611,
0.03214333578944206,
0.022362450137734413,
0.02868521213531494,
0.03747367486357689,
-0.013848853297531605,
0.01262876857072115,
0.01917218789458275,
0.050562068819999695,
-0.03873104974627495,
0.06626532971858978,
-0.012510129250586033,
-0.027139650657773018,
0.04113888368010521,
0.01924900896847248,
-0.0724184438586235,
0.055164143443107605,
0.007283222395926714,
-0.09123961627483368,
0.0261596217751503,
0.012862174771726131,
-0.04175560921430588,
-0.10409167408943176,
0.049548275768756866,
-0.026391761377453804,
-0.009091544896364212,
0.035866208374500275,
0.012729606591165066,
-0.12759090960025787,
0.004093904979526997,
0.11889903992414474,
-0.07418325543403625,
-0.04649359732866287,
0.0024380763061344624,
0.04228862375020981,
-0.11901560425758362,
-0.0394095703959465,
0.040799468755722046,
0.015984946861863136,
0.04737585410475731,
-0.02450716868042946,
0.02766239456832409,
0.06644042581319809,
-0.06122887507081032,
0.04245886579155922,
0.023366490378975868,
0.007819345220923424,
-0.0166803989559412,
0.013213155791163445,
-0.0350181944668293,
-0.007435524836182594,
0.05473315715789795,
-0.048327941447496414,
0.046282511204481125,
-0.07278892397880554,
-0.13611997663974762,
0.018312446773052216,
-0.012413389049470425,
0.015218883752822876,
0.09745492786169052,
-2.6672527016684267e-33,
-0.012450712732970715,
-0.018089348450303078,
-0.020068323239684105,
-0.08918936550617218,
-0.06028931960463524,
-0.02251083217561245,
0.09684811532497406,
-0.09030917286872864,
0.056620821356773376,
0.034366220235824585,
0.06017579510807991,
-0.010262019000947475,
-0.027246659621596336,
0.04778843745589256,
-0.01163480430841446,
-0.0851074829697609,
-0.0237312950193882,
-0.0665336549282074,
-0.01564725860953331,
-0.02052684687077999,
0.013715938664972782,
-0.05282193422317505,
-0.005012199282646179,
-0.030301902443170547,
-0.006512703839689493,
0.02600797824561596,
0.06518235802650452,
-0.0815039798617363,
-0.018345028162002563,
0.02713601663708687,
0.010545662604272366,
-0.04489579424262047,
0.0284977238625288,
0.054964639246463776,
-0.006697653327137232,
-0.05918915197253227,
-0.024074368178844452,
-0.08951491862535477,
0.0860653892159462,
-0.06520449370145798,
0.05008251219987869,
0.040459830313920975,
0.08004102110862732,
0.02571169286966324,
-0.015328285284340382,
0.06364291161298752,
-0.018536105751991272,
0.11286351829767227,
-0.04497769847512245,
0.035631511360406876,
0.026211896911263466,
-0.08697570115327835,
-0.02346617542207241,
-0.020172443240880966,
0.10825508087873459,
-0.07833356410264969,
0.037623949348926544,
-0.030825471505522728,
0.05256346985697746,
-0.032483700662851334,
0.01949908956885338,
-0.06853670626878738,
0.06023816764354706,
0.04301181063055992,
-0.051948852837085724,
0.016019335016608238,
-0.03036806359887123,
-0.10352770239114761,
0.05577026680111885,
-0.031609874218702316,
0.012801769189536572,
-0.024307694286108017,
-0.04034747928380966,
-0.08200227469205856,
-0.03875129297375679,
-0.007126738782972097,
-0.019650030881166458,
0.11798225343227386,
0.09415163844823837,
-0.07609306275844574,
-0.03877418115735054,
0.05753041058778763,
-0.028554514050483704,
-0.018995890393853188,
-0.06164528802037239,
0.04554038867354393,
0.035503726452589035,
0.005593498703092337,
0.044384267181158066,
-0.03955332562327385,
0.08108062297105789,
0.029177820309996605,
0.014367626048624516,
-0.0411229282617569,
0.06652279198169708,
6.976161655039698e-32,
-0.011964548379182816,
0.05504576116800308,
0.0072175427339971066,
0.016307786107063293,
-0.014877400361001492,
0.009854757227003574,
0.01499865297228098,
0.05037344992160797,
-0.0010520901996642351,
-0.009917985647916794,
0.09100323170423508,
0.043000198900699615,
-0.1277202069759369,
-0.013468782417476177,
-0.017305394634604454,
0.014578931033611298,
-0.0352606326341629,
0.015471488237380981,
0.036706000566482544,
0.004024541936814785,
-0.039440736174583435,
0.0017651303205639124,
-0.03794737905263901,
0.10574381798505783,
0.019826149567961693,
0.02052396908402443,
-0.03026791661977768,
-0.046660732477903366,
0.01704203523695469,
0.011596547439694405,
0.051048506051301956,
-0.03159477561712265,
-0.030180424451828003,
-0.05443008616566658,
0.12011146545410156,
0.07414360344409943,
-0.13192370533943176,
0.04821927472949028,
0.03641802445054054,
-0.005603542551398277,
-0.015191447921097279,
0.008474176749587059,
0.007344662211835384,
-0.06445126235485077,
-0.01620412990450859,
-0.05699532479047775,
0.024860277771949768,
-0.054219454526901245,
0.034711092710494995,
-0.03715703636407852,
-0.04641979932785034,
0.12630286812782288,
-0.04102012887597084,
0.054866258054971695,
-0.003069482510909438,
-0.031300466507673264,
-0.011619177646934986,
-0.02521730586886406,
0.0042430018074810505,
-0.06012360006570816,
-0.0015336889773607254,
0.03957294672727585,
-0.06858014315366745,
-0.00534778693690896
] |
46
Chapter 3
3.4 Linear inequalities
You can solve linear inequalities using similar methods to those for solving linear equations.
β The solution of an inequality is the set o
f all real numbers x that make the inequality true.
Example 7
Find the set of values of x for which:
a 5x
+ 9 > x + 20 b 12 β 3
x < 27
c 3(x
β 5) > 5 β 2(x β 8)
a 5x + 9 > x + 20
4x + 9 > 20
4x > 11
x > 2.75
b 12 β
3x < 27
β3x < 15
x > β5
c 3(x
β 5) > 5 β 2( x β 8)
3x β 15 > 5 β 2x + 16
5x > 5 + 16 + 15
5x > 36
x > 7.2Subtract 12 from both sides.
Divide both sides by β3. (You therefore need to
turn round the inequality sign.)
In set notation {x : x >
β5}.
Multiply out (note: β2 Γ β8 = +16).Rearrange to get x > β¦In set notation {x
: x >
7.2}.
β6 β4 β2 0 2 4 6
Here the solution sets are x < β1 or x > 3.
β6 β4 β2 0 2 4 6Here there is no overlap and the two inequalities have
to be written separately as x < β1 or x > 3.β is used for < and > and means the end value is not included.
β is used for < and > and means the end value is
included.
These are the only real values that satisfy both
equalities simultaneously so the solution is β2 < x < 4.You may sometimes need to find the set of
values for which two inequalities are true
together. Number lines can be useful to find your solution.
For example, in the number line below the
solution set is x > β2 and x < 4. In se t notation
x > β2 and x < 4 is written { x : β2 < x < 4}
or alternatively { x : x > β2} β { x : x < 4}
x < β1 or x > 3 is written { x : x < β1} β { x : x > 3}Notation You c an write the solution to this
inequality using set notation as { x : x >
2.75}.
This means the set of all values x for which x is
greater than or equal to 2.75.Notation
Rearrange to get x > β¦
|
[
0.016399066895246506,
0.06043170019984245,
0.09353082627058029,
-0.05480608716607094,
0.019891047850251198,
0.00316721317358315,
-0.0984192043542862,
0.034346237778663635,
-0.12316211313009262,
0.06643901765346527,
-0.012915395200252533,
-0.06189008429646492,
0.019199756905436516,
0.009743538685142994,
0.007073414511978626,
0.006102447863668203,
-0.012889093719422817,
-0.004862567875534296,
-0.06110592186450958,
0.0046461522579193115,
-0.013823186978697777,
-0.07846438139677048,
-0.07523345947265625,
0.019224099814891815,
0.08421443402767181,
-0.10133003443479538,
-0.07107996940612793,
-0.049028635025024414,
-0.025755297392606735,
-0.010282795876264572,
-0.014749298803508282,
-0.0830719843506813,
0.10190414637327194,
-0.08932285010814667,
-0.010985750705003738,
0.051728591322898865,
-0.009434113278985023,
0.0537090003490448,
0.0031468451488763094,
0.009891227819025517,
-0.005161150824278593,
0.03579326346516609,
-0.001019989955238998,
-0.024991802871227264,
0.01034675445407629,
-0.02632138878107071,
0.05359724164009094,
-0.04563656076788902,
0.04229434207081795,
0.03037666156888008,
0.008263710886240005,
-0.05196226015686989,
0.053751252591609955,
0.08114372938871384,
-0.06132042407989502,
-0.06597769260406494,
0.019505245611071587,
-0.05914400517940521,
-0.0353073924779892,
0.006827244535088539,
-0.04459521546959877,
-0.024049492552876472,
0.0137092350050807,
0.04730891063809395,
0.07085859775543213,
0.06888306885957718,
0.01454836130142212,
0.01257892232388258,
-0.0198442954570055,
0.020917506888508797,
-0.08430630713701248,
0.02875816635787487,
0.022034408524632454,
-0.02133692242205143,
0.016293969005346298,
0.06517622619867325,
-0.05795624107122421,
-0.11044226586818695,
0.024772293865680695,
0.0067419083788990974,
0.011506027542054653,
0.0346253477036953,
0.06278162449598312,
0.013100548647344112,
0.00929141603410244,
-0.03147411346435547,
0.17212152481079102,
0.06321115046739578,
-0.03457588702440262,
0.06761377304792404,
0.01887519657611847,
-0.010742646642029285,
0.009794393554329872,
-0.0021135953720659018,
0.005438109394162893,
-0.11590657383203506,
0.019577745348215103,
0.01336018368601799,
-0.0076932585798203945,
0.007473694626241922,
-0.004478415474295616,
-0.08897930383682251,
0.029038801789283752,
-0.012417987920343876,
-0.0007769098156131804,
0.00008578126289648935,
-0.0401691310107708,
-0.029321124777197838,
0.02634541690349579,
-0.04595085605978966,
-0.03125576674938202,
-0.11117909848690033,
0.022534146904945374,
0.06897228211164474,
0.04407928138971329,
-0.04849633574485779,
0.014316127635538578,
0.04230588302016258,
-0.002294322242960334,
-0.09074917435646057,
-0.06281061470508575,
-0.04053815081715584,
-0.001866075093857944,
-0.0360911600291729,
-0.03259597718715668,
0.0014844076940789819,
0.06600894778966904,
0.006550328340381384,
-0.09563151746988297,
-0.11118990927934647,
0.021966001018881798,
-0.018369778990745544,
-0.02866998128592968,
-0.04771172255277634,
-0.03128180652856827,
-0.0714099109172821,
-0.03152639418840408,
0.026780229061841965,
-0.029716357588768005,
-0.04082239791750908,
0.015290223062038422,
0.04090430215001106,
0.11246117204427719,
0.08753033727407455,
0.017699498683214188,
0.003822924103587866,
-0.015980971977114677,
-0.024632062762975693,
-0.031830113381147385,
-0.12334350496530533,
-0.028484545648097992,
0.02305036224424839,
0.05810747668147087,
0.07362508028745651,
-0.12094047665596008,
-0.05858072265982628,
-0.04136110842227936,
0.01377646904438734,
-0.0032159320544451475,
0.05126750096678734,
-0.06084267422556877,
0.057944320142269135,
0.026209769770503044,
0.061291735619306564,
-0.03849881887435913,
0.021829314529895782,
-0.02375025302171707,
0.04815825819969177,
-0.06097972020506859,
-0.07015825062990189,
0.10864285379648209,
0.016858920454978943,
0.03706640005111694,
0.1205618679523468,
-0.06894619762897491,
0.039918508380651474,
0.03443760797381401,
0.03161225467920303,
-0.0680684819817543,
0.0866774320602417,
0.03254036605358124,
-0.04494558274745941,
0.0039224643260240555,
-0.0050384956412017345,
-0.03193041682243347,
0.03326144069433212,
0.028261004015803337,
-0.06035332381725311,
-0.01250283233821392,
-0.10090429335832596,
0.021150782704353333,
-0.08998037874698639,
-0.01056008879095316,
0.07425623387098312,
-0.061843715608119965,
-0.07136748731136322,
0.037553977221250534,
0.00972495786845684,
-0.012871582992374897,
0.017816096544265747,
-0.013776920735836029,
-0.010618596337735653,
0.01436175499111414,
0.04380333423614502,
0.0006523181800730526,
0.0267442986369133,
-0.05670367553830147,
-0.024279609322547913,
0.07091321796178818,
0.015329425223171711,
-0.10024948418140411,
0.012397016398608685,
-0.06597583740949631,
-0.007583573926240206,
0.02539285272359848,
-0.06628027558326721,
0.0846521407365799,
-0.029656976461410522,
-0.029397331178188324,
0.025165127590298653,
-0.029699141159653664,
-0.03304062411189079,
0.047648001462221146,
-7.643164920750303e-33,
-0.06610912829637527,
0.0495787039399147,
-0.056740470230579376,
-0.03651224449276924,
0.030469495803117752,
0.014782963320612907,
0.04372745752334595,
-0.11175613105297089,
0.02022688277065754,
-0.036800529807806015,
0.02629033476114273,
-0.002801454858854413,
-0.038706663995981216,
-0.030276324599981308,
-0.006253386847674847,
-0.07020626962184906,
-0.026985548436641693,
-0.008312861435115337,
0.04511642828583717,
-0.04947209358215332,
0.032983869314193726,
-0.04846280440688133,
-0.062207482755184174,
0.0226715337485075,
0.04186408221721649,
-0.011221335269510746,
0.05128660425543785,
0.047840237617492676,
-0.01041216030716896,
-0.004439949057996273,
-0.05402844026684761,
0.027682138606905937,
0.062307972460985184,
0.044922709465026855,
-0.008078656159341335,
-0.12444225698709488,
0.020392481237649918,
-0.04301777482032776,
0.0399455763399601,
-0.06578880548477173,
-0.012791489250957966,
0.03170626610517502,
0.06115991622209549,
0.027258172631263733,
-0.012219361960887909,
0.023076049983501434,
0.05996531993150711,
0.04988286271691322,
0.003997871186584234,
0.08265232294797897,
0.03353043645620346,
0.009097193367779255,
-0.020158642902970314,
-0.006141249090433121,
0.004087988752871752,
-0.028879277408123016,
0.021890806034207344,
0.005891308654099703,
-0.022145507857203484,
0.02032107673585415,
-0.005775946658104658,
-0.08836696296930313,
0.13529092073440552,
0.02440149150788784,
-0.08560363948345184,
0.07747715711593628,
-0.039394836872816086,
-0.0509926974773407,
0.01729213446378708,
0.0035470493603497744,
-0.007786812726408243,
-0.017087195068597794,
0.017393846064805984,
-0.044305890798568726,
0.011899460107088089,
0.04911723732948303,
-0.032038308680057526,
0.05913389101624489,
0.003996117971837521,
0.029270866885781288,
-0.027785522863268852,
0.024450138211250305,
0.06599712371826172,
-0.06772208213806152,
-0.12102818489074707,
0.04751545935869217,
0.06506878137588501,
0.05517478659749031,
0.032311342656612396,
-0.011567603796720505,
0.052177369594573975,
0.031257640570402145,
-0.05934995040297508,
-0.0356169268488884,
0.1378023624420166,
6.834611564049556e-32,
0.005206648260354996,
-0.02103971131145954,
-0.07861440628767014,
0.052969880402088165,
0.03376222774386406,
-0.010410926304757595,
-0.011059253476560116,
0.05217303708195686,
0.041851941496133804,
-0.059341683983802795,
0.044790834188461304,
-0.008510884828865528,
-0.07337982952594757,
0.025168871507048607,
-0.03412524610757828,
-0.024259202182292938,
-0.078652024269104,
-0.005795312579721212,
0.04219469055533409,
-0.012254700995981693,
0.0003044566255994141,
0.020065253600478172,
-0.018224388360977173,
0.04481741413474083,
0.04976169764995575,
0.0044933948665857315,
-0.09936416149139404,
0.08208242803812027,
0.04632017761468887,
0.04490572586655617,
0.031977180391550064,
0.0010624458082020283,
0.040740810334682465,
-0.04615858197212219,
0.14613395929336548,
0.0593181774020195,
-0.011798552237451077,
0.007032547611743212,
-0.029804522171616554,
-0.04525231942534447,
0.04572214186191559,
-0.014528323896229267,
-0.03193887695670128,
-0.06459738314151764,
-0.043882083147764206,
-0.06552698463201523,
0.05473003536462784,
-0.03692835569381714,
0.08382134884595871,
-0.04195525124669075,
-0.011734196916222572,
0.12881344556808472,
0.005754492245614529,
0.0587981678545475,
-0.019621828570961952,
-0.03073190338909626,
-0.04015656188130379,
-0.03775204345583916,
-0.021600984036922455,
-0.1309121549129486,
0.05624660104513168,
0.021577855572104454,
-0.03364258632063866,
0.016089102253317833
] |
47
Equations and inequalities
Example 8
Find the set of values of x for which:
a 3x
β 5 < x + 8 and 5x > x β 8
b x β
5 > 1 β x or 15 β 3x > 5 + 2x.
c 4x
+ 7 > 3 and 17 < 11 + 2x.
a 3x β 5 < x + 8 5x > x β 8
2x β 5 < 8 4x >
β 8
2x < 13 x >
β2
x < 6.5
β/four.ss01 β2 0 /four.ss01 2 6 8
x < 6.5
x > β2
So the required set of values is β 2 < x < 6.5.
b x β
5 > 1 β x 15 β
3x > 5 + 2x
2x β 5 > 1 10 β
3x > 2x
2x > 6 10 >
5x
x > 3 2 >
x
x <
2
x < 3
x > 2β/four.ss01 β2 0 /four.ss01 2 6 8
The solution is x > 3 or x < 2.Draw a number line to illustrate the two
inequalities.
The two sets of values overlap (intersect) where
β2 < x < 6.5.
Notice here how this is written when x lies
between two values.
In set notation this can be written as
{x : β2 < x < 6.5}.
Draw a number line. Note that there is no overlap between the two sets of values.
In set notation this can be written as
{x : x < 2} β {x : x > 3}.
1 Find the set of va lues of x for which:
a 2x
β 3 < 5 b 5x
+ 4 > 39
c 6x
β 3 > 2x + 7 d 5x
+ 6 < β12 β x
e 15 β
x > 4 f 21 β 2
x > 8 + 3x
g 1 +
x < 25 + 3x h 7x
β 7 < 7 β 7x
i 5 β 0.5
x > 1 j 5x
+ 4 > 12 β 2xExercise 3D
|
[
0.01421315222978592,
0.07041201740503311,
0.09081154316663742,
-0.05478221923112869,
0.02662539854645729,
0.022491853684186935,
-0.08873472362756729,
0.02521478198468685,
-0.12184783071279526,
0.033061299473047256,
-0.015564900822937489,
-0.07961013913154602,
0.048516351729631424,
0.018879882991313934,
0.01766127720475197,
0.003929598722606897,
0.006510355044156313,
-0.027941854670643806,
-0.06370101124048233,
-0.04347610101103783,
-0.014684421010315418,
-0.09327496588230133,
-0.030122321099042892,
-0.01908838376402855,
0.06116552650928497,
-0.09721288830041885,
-0.026662515476346016,
-0.024619461968541145,
0.005381477065384388,
-0.02629481628537178,
-0.01982927694916725,
-0.045586373656988144,
0.1112443283200264,
-0.05220470204949379,
0.044075287878513336,
0.018257204443216324,
0.08037793636322021,
0.041206661611795425,
-0.010891968384385109,
-0.015295947901904583,
-0.025653144344687462,
0.008210965432226658,
0.000978525960817933,
0.01367594487965107,
-0.0381927415728569,
0.006135666277259588,
0.050839874893426895,
-0.037755079567432404,
0.004225964657962322,
-0.013105930760502815,
0.05807402357459068,
-0.0007396807195618749,
0.04215611517429352,
0.0947035700082779,
-0.04172065109014511,
-0.07131240516901016,
-0.002828909084200859,
-0.0420842207968235,
-0.07149811834096909,
-0.015823446214199066,
-0.03861657902598381,
-0.03490697965025902,
-0.010798395611345768,
0.036707475781440735,
0.03517737612128258,
0.09650242328643799,
0.02291707880795002,
-0.03199058771133423,
-0.006324721500277519,
0.06384368985891342,
-0.153402179479599,
0.05449317395687103,
-0.040237974375486374,
-0.0867251604795456,
0.015176218934357166,
-0.015416966751217842,
-0.060513705015182495,
-0.05047206953167915,
-0.0007303459569811821,
-0.05288432538509369,
-0.009463779628276825,
0.010218601673841476,
0.05059012025594711,
0.021693766117095947,
-0.03783737123012543,
-0.06843919306993484,
0.11705373227596283,
0.07409148663282394,
-0.01935512199997902,
-0.01412668451666832,
0.04994777590036392,
-0.01116151548922062,
0.013127506710588932,
-0.02437601424753666,
0.055694106966257095,
-0.1284925639629364,
0.009602071717381477,
-0.054773878306150436,
-0.02514095976948738,
0.01710166223347187,
0.022585634142160416,
-0.0836709663271904,
0.06661149114370346,
-0.00012345371942501515,
-0.0190797857940197,
-0.015248377807438374,
-0.03288574889302254,
-0.0480029322206974,
0.004415158182382584,
-0.09285713732242584,
-0.03942808508872986,
-0.07027224451303482,
-0.0027843245770782232,
0.053539253771305084,
-0.021978113800287247,
-0.06185686215758324,
0.04355134814977646,
0.048645664006471634,
0.05707607418298721,
-0.014929378405213356,
-0.036841221153736115,
-0.033865686506032944,
0.020694801583886147,
-0.04985525831580162,
-0.04620848596096039,
-0.008301382884383202,
0.034190256148576736,
0.03969959914684296,
-0.019991355016827583,
-0.12544193863868713,
-0.005697184707969427,
-0.026630239561200142,
-0.018678709864616394,
-0.02482924610376358,
-0.0025148370768874884,
-0.11850392073392868,
-0.04484415054321289,
0.06030973419547081,
-0.019039787352085114,
-0.022518623620271683,
-0.0037146923132240772,
0.05569121241569519,
0.07360804080963135,
0.04393090307712555,
0.03294089809060097,
0.0012732275063171983,
-0.09563281387090683,
-0.03494200482964516,
-0.04622630402445793,
-0.03705422207713127,
-0.031053300946950912,
0.003495125798508525,
0.05846850946545601,
0.09199082106351852,
-0.051508452743291855,
-0.05910630524158478,
-0.0524434968829155,
-0.010150004178285599,
-0.00775530468672514,
0.03542855381965637,
-0.07560942322015762,
0.047140415757894516,
0.015452354215085506,
0.11020804941654205,
0.03350881114602089,
-0.00682303449138999,
0.02475370094180107,
0.038724567741155624,
-0.05390792340040207,
-0.03576279804110527,
0.045559946447610855,
0.02576749213039875,
0.018320737406611443,
0.043181002140045166,
-0.05888798087835312,
-0.005380702670663595,
0.08303843438625336,
0.04670995473861694,
-0.0766635611653328,
0.10904645174741745,
0.0577719546854496,
-0.053927842527627945,
-0.03794540464878082,
-0.0022998482454568148,
-0.06675530970096588,
0.012435086071491241,
-0.00783519633114338,
-0.018530838191509247,
0.004905458074063063,
-0.01521677803248167,
0.024674877524375916,
-0.14443251490592957,
0.03735297918319702,
0.06581717729568481,
-0.07201655954122543,
-0.0329122431576252,
0.008879668079316616,
-0.0031911765690892935,
-0.05957057327032089,
-0.0015352857299149036,
0.04289357364177704,
-0.003099215216934681,
0.08635236322879791,
0.05819683521986008,
-0.02647782862186432,
0.01952630840241909,
-0.08232953399419785,
0.011094427667558193,
0.07869547605514526,
0.011216576211154461,
-0.035934608429670334,
0.039900947362184525,
-0.08108970522880554,
-0.017680155113339424,
0.004289655946195126,
-0.06467290222644806,
0.09866680204868317,
-0.08805080503225327,
-0.023383919149637222,
-0.03801232576370239,
-0.01698950305581093,
-0.040804289281368256,
0.009156402200460434,
-6.314298679188446e-33,
-0.08513224869966507,
0.012498450465500355,
-0.0765032172203064,
-0.07379963248968124,
-0.012532423250377178,
0.008702688850462437,
0.03804782032966614,
-0.035175688564777374,
0.055465277284383774,
-0.017663752660155296,
0.08769286423921585,
0.021979883313179016,
-0.023587960749864578,
0.008276054635643959,
-0.05993478000164032,
-0.06460227072238922,
-0.048921361565589905,
0.04957059025764465,
0.06721396744251251,
-0.015457694418728352,
0.03514183685183525,
0.007643990218639374,
-0.004085136577486992,
0.010563828982412815,
0.02010442689061165,
0.04818720370531082,
0.0363248810172081,
-0.028326358646154404,
0.014740058220922947,
-0.0176150631159544,
0.012457547709345818,
-0.007745719514787197,
0.06780306994915009,
0.014276537112891674,
0.007244830019772053,
-0.13534896075725555,
0.008483598940074444,
-0.030845392495393753,
0.053856268525123596,
-0.017654992640018463,
0.01626680977642536,
-0.01830543763935566,
0.09158466011285782,
0.04149077832698822,
0.0019490162376314402,
0.007330575957894325,
0.02445688471198082,
0.0913442000746727,
0.04562638700008392,
0.010274030268192291,
0.05048073083162308,
-0.04315947741270065,
-0.0659828707575798,
0.0024761976674199104,
0.08060114830732346,
-0.026173315942287445,
0.017329908907413483,
-0.018130648881196976,
0.029894258826971054,
-0.01944642700254917,
0.010050320997834206,
-0.0779731348156929,
0.08453652262687683,
0.025069545954465866,
-0.06762662529945374,
0.0377623476088047,
-0.022283663973212242,
-0.08628436923027039,
0.018820028752088547,
-0.021623779088258743,
0.05662240460515022,
-0.0631943792104721,
-0.014770588837563992,
-0.0955621525645256,
-0.04391183331608772,
0.04297216981649399,
0.006536646746098995,
0.07373178750276566,
0.04511208459734917,
0.00627446174621582,
-0.09405982494354248,
0.06917797029018402,
0.02918749861419201,
0.010272443294525146,
-0.0827309861779213,
0.03058226779103279,
0.11036349087953568,
0.0396689809858799,
0.06472060084342957,
0.006213840562850237,
0.038086969405412674,
-0.007564533967524767,
0.003060889197513461,
-0.04139255732297897,
0.09984643012285233,
6.706212316109237e-32,
-0.011613992042839527,
-0.009092427790164948,
-0.06022176519036293,
0.04806288331747055,
0.004071356263011694,
-0.0336044616997242,
-0.10233639925718307,
0.0010290135396644473,
0.04925708845257759,
-0.0598837211728096,
0.10049823671579361,
0.06792599707841873,
-0.1202278882265091,
0.023197785019874573,
-0.032078295946121216,
0.0030607508961111307,
-0.051949772983789444,
0.030367165803909302,
0.03231123834848404,
0.014986386522650719,
-0.0017929681343957782,
0.0883876159787178,
0.00462930416688323,
0.06122796609997749,
0.04482077807188034,
-0.005444992333650589,
-0.0994497761130333,
-0.009992252103984356,
0.05080395191907883,
0.0792577862739563,
0.0332692451775074,
0.013263450004160404,
-0.02597058191895485,
-0.014444740489125252,
0.08028896152973175,
0.04245069995522499,
-0.032626837491989136,
0.0007286987383849919,
0.017783114686608315,
0.026614166796207428,
0.04416266083717346,
-0.0009574968717060983,
-0.032038282603025436,
-0.05454910546541214,
-0.0036218450404703617,
-0.09367166459560394,
0.025446755811572075,
-0.004627718590199947,
0.031614191830158234,
-0.07418274879455566,
-0.07886350154876709,
0.1332721710205078,
0.014990980736911297,
0.037652477622032166,
0.0195977333933115,
-0.03419660031795502,
0.012889307923614979,
-0.024524778127670288,
-0.00286694779060781,
-0.09155963361263275,
0.054221730679273605,
0.06829799711704254,
-0.06754077225923538,
0.04618428647518158
] |
48
Chapter 3
3.5 Quadratic inequalities
β To solve a quadratic inequality:
β’ Rearr
ange so that the right-hand side of the inequality is 0
β’ Solve the corresponding quadratic equation to find the critical values
β’ Sketch the graph of the quadratic function
β’ Use your sketch to find the required set of values.
The sketch shows the graph of f(x) = x2 β 4x β 5
= (x + 1)(x β 5)
β1 5y
x OThe solutions to the quadratic inequality
x2 β 4x β 5 > 0 are the x-values when
the curve is above the x-axis (the darker
part of the curve). This is when x < β1 or x > 5. In set notation the solution is {x : x < β1} β {x : x > 5}.
The solutions to the quadratic inequality x
2 β 4x β 5 < 0 are the x-values when
the curve is below the x-axis (the lighter part of the curve). This is when x > β1 and x < 5 or β1 < x < 5. In set notation the solution is {x : β1 < x < 5}.The solutions to f(x) = 0 are x = β1 and x = 5. These are called the critical values.A = { x : 3x + 5 > 2} B = { x : x __ 2 + 1 < 3 } C = { x : 11 < 2x β 1}
Given that A β ( B β C ) = {x :
p < x < q } β {x : x > r }, find the values of p , q and r .Challenge2 Find the set of va lues of x for which:
a 2(x
β 3) > 0 b 8(1 β
x) > x β 1 c 3(x
+ 7) < 8 β x
d 2(x
β 3) β (x + 12) < 0 e 1 + 11(2
β x) < 10(x β 4) f 2(x
β 5) > 3(4 β x)
g 12x
β 3(x β 3) < 45 h x β
2(5 + 2x) < 11 i x(x
β 4) > x2 + 2
j x(5 β
x) > 3 + x β x2 k 3x + 2x(x β 3) < 2(5 + x2) l x(2x β 5) < 4x(
x + 3) ________ 2 β 9
3 Use set notation to describe the set of v
alues of x for which:
a 3(x
β 2) > x β 4 and 4x + 12 > 2x + 17
b 2x
β 5 < x β 1 and 7(x + 1) > 23 β x
c 2x
β 3 > 2 and 3(x + 2) < 12 + x
d 15 β
x < 2(11 β x) and 5(3x β 1) > 12x + 19
e 3x
+ 8 < 20 and 2(3x β 7) > x + 6
f 5x
+ 3 < 9 or 5(2x + 1) > 27
g 4(3x
+ 7) < 20 or 2(3x β 5) > 7 β 6
x ______ 2
|
[
0.07065148651599884,
0.09809315949678421,
0.08456576615571976,
-0.06121457368135452,
-0.03244809806346893,
0.038358449935913086,
-0.0027155124116688967,
0.060875989496707916,
-0.07765325158834457,
0.07579487562179565,
-0.03589198365807533,
-0.06194010004401207,
-0.009178942069411278,
0.1039406880736351,
-0.04848784953355789,
0.02158941701054573,
-0.0483858585357666,
-0.05399315431714058,
-0.05637722462415695,
-0.0494694821536541,
-0.013033241033554077,
-0.10256209969520569,
-0.03312110900878906,
-0.06661196798086166,
0.08033887296915054,
-0.09368915855884552,
-0.01993410289287567,
-0.11502328515052795,
-0.006233503110706806,
-0.0003995191946160048,
0.05358361825346947,
-0.057006172835826874,
-0.0036671028938144445,
-0.04105796664953232,
0.025779878720641136,
-0.0017185016768053174,
0.04393686354160309,
0.022571006789803505,
0.05960690230131149,
-0.021467464044690132,
0.03354496881365776,
0.006717017386108637,
-0.08943149447441101,
-0.04185082018375397,
-0.030862262472510338,
-0.014661439694464207,
-0.021834449842572212,
-0.08138152956962585,
0.07685719430446625,
0.03532518446445465,
-0.005906559061259031,
0.017891133204102516,
-0.009560083039104939,
0.056031398475170135,
-0.03950386494398117,
-0.044362619519233704,
0.0547395795583725,
-0.09508991986513138,
-0.051732346415519714,
0.08761262148618698,
0.05003717541694641,
0.003850995795801282,
-0.037612564861774445,
0.04539060220122337,
0.03836900368332863,
0.013973102904856205,
0.06494475901126862,
0.02229798026382923,
-0.036418367177248,
0.09959975630044937,
-0.09286721795797348,
-0.07063250243663788,
0.035709820687770844,
-0.06517382711172104,
0.04798653721809387,
0.01384709868580103,
-0.03749934211373329,
-0.020826173946261406,
-0.007302338723093271,
-0.06538441777229309,
0.008883554488420486,
0.03387035056948662,
0.058496277779340744,
0.012294127605855465,
-0.06876733154058456,
-0.07964970171451569,
0.06664874404668808,
0.05603613331913948,
0.0037774115335196257,
0.05294394865632057,
0.010290813632309437,
0.00942473765462637,
0.016904395073652267,
-0.023854311555624008,
0.031037045642733574,
-0.10014566034078598,
0.08393292874097824,
-0.009976356290280819,
-0.04382344335317612,
0.019445639103651047,
-0.019915543496608734,
-0.06543650478124619,
0.004504152107983828,
0.02295316383242607,
-0.029184069484472275,
0.03230423852801323,
0.03307485207915306,
0.022906960919499397,
-0.0031053652055561543,
-0.08129662275314331,
-0.03843935579061508,
-0.08384615182876587,
0.024613777175545692,
0.09066170454025269,
0.03451243042945862,
-0.0010189693421125412,
0.060973428189754486,
0.00118797249160707,
-0.013162201270461082,
-0.055264703929424286,
-0.014018076471984386,
-0.04179685562849045,
0.000004858380179939559,
-0.023253172636032104,
-0.0539412684738636,
-0.003260923083871603,
0.03173268586397171,
0.013881819322705269,
0.026326442137360573,
-0.05545517057180405,
0.01622130535542965,
-0.03384200111031532,
-0.011214496567845345,
-0.07948043197393417,
-0.016576318070292473,
-0.10766083002090454,
-0.019628731533885002,
0.09860439598560333,
-0.0025908935349434614,
-0.03887975588440895,
-0.02675904519855976,
0.039493780583143234,
0.05573342740535736,
0.04910234361886978,
0.013736008666455746,
-0.04352443665266037,
-0.02809709869325161,
-0.00987556204199791,
0.012286465615034103,
-0.060408949851989746,
-0.022921212017536163,
-0.01121043972671032,
0.04989885911345482,
0.13890425860881805,
0.03601166978478432,
-0.09694622457027435,
-0.07275762408971786,
0.03441168740391731,
-0.0009870716603472829,
0.003745569847524166,
-0.06478182226419449,
-0.005044632591307163,
-0.022388743236660957,
0.01550925150513649,
-0.0440465547144413,
0.015121608041226864,
0.013775837607681751,
0.047305792570114136,
0.037568096071481705,
-0.11888141930103302,
0.07268298417329788,
0.028993967920541763,
0.030210893601179123,
0.07720955461263657,
-0.04740944877266884,
0.012554211542010307,
0.07347515970468521,
0.03763609752058983,
0.013561452738940716,
0.03848922252655029,
0.03728323057293892,
-0.037281740456819534,
-0.0284352358430624,
-0.010221708565950394,
0.05670371279120445,
-0.042547691613435745,
0.0003588291583582759,
-0.09818388521671295,
0.017665473744273186,
-0.083128422498703,
0.01732412911951542,
-0.07895666360855103,
0.019123131409287453,
0.06525691598653793,
-0.09007364511489868,
-0.0013067666441202164,
0.050816189497709274,
0.05248209089040756,
-0.09315445274114609,
0.019487984478473663,
0.038784921169281006,
-0.005485575646162033,
0.13306282460689545,
0.06728846579790115,
-0.007961669936776161,
-0.03434617817401886,
-0.043121226131916046,
0.01401848066598177,
0.0166451558470726,
-0.008354620076715946,
-0.037614334374666214,
0.032017070800065994,
-0.05461164191365242,
-0.016881262883543968,
0.023756764829158783,
-0.06023993715643883,
0.09752587974071503,
0.010136015713214874,
-0.059371788054704666,
0.01932266168296337,
-0.1067105382680893,
0.0597122386097908,
0.024267729371786118,
-2.4164585941975916e-33,
-0.016444258391857147,
-0.021646620705723763,
-0.0801955908536911,
-0.0031028911471366882,
-0.031962234526872635,
-0.014271785505115986,
0.06609687954187393,
-0.07004369795322418,
0.029559368267655373,
0.08994966000318527,
-0.0009009168134070933,
0.022891608998179436,
-0.026808321475982666,
0.06256411224603653,
-0.03810479864478111,
-0.04113555699586868,
-0.03362688049674034,
0.06299936771392822,
-0.038761354982852936,
-0.09098096191883087,
0.0845034047961235,
-0.04866037890315056,
0.004022838082164526,
-0.03394879773259163,
0.038044992834329605,
0.06051408872008324,
0.0854690745472908,
-0.045021966099739075,
0.03857988864183426,
0.0032957298681139946,
-0.05522691830992699,
-0.008102171123027802,
0.006731869652867317,
0.006497751455754042,
0.042441997677087784,
-0.08264848589897156,
0.002143530175089836,
-0.13225342333316803,
0.12761518359184265,
-0.04978358373045921,
0.04493061453104019,
0.016032841056585312,
0.10277600586414337,
0.010356339626014233,
0.03057612106204033,
0.04148022085428238,
0.02806258015334606,
0.019180770963430405,
-0.02885490655899048,
0.045843783766031265,
0.01850380375981331,
-0.023819953203201294,
0.060278043150901794,
0.010777294635772705,
0.0457821823656559,
-0.05201398581266403,
0.05609942972660065,
0.06204644963145256,
-0.005648917984217405,
0.060995329171419144,
0.0039820424281060696,
-0.08638593554496765,
0.02015852928161621,
0.002751097548753023,
-0.06564919650554657,
-0.00842322874814272,
-0.004906792193651199,
-0.004982704762369394,
0.038384102284908295,
0.04049818217754364,
-0.03313795477151871,
-0.019532110542058945,
0.023399468511343002,
-0.06423412263393402,
-0.04764208570122719,
0.01799752190709114,
-0.04969785362482071,
0.04806225001811981,
0.05663786828517914,
-0.06608578562736511,
-0.027745293453335762,
0.06723586469888687,
-0.0508107915520668,
-0.03998548537492752,
-0.0771666169166565,
0.028367485851049423,
-0.0038880594074726105,
0.08404338359832764,
0.07807634025812149,
0.0030246409587562084,
-0.011067759245634079,
-0.01906587928533554,
-0.09137176722288132,
-0.010929538868367672,
-0.025284353643655777,
5.658517711108349e-32,
-0.05692296475172043,
-0.004218041431158781,
-0.009131846949458122,
0.03996775671839714,
-0.007727424148470163,
0.004655114840716124,
0.06469997763633728,
-0.02629254385828972,
0.040556661784648895,
-0.03527267277240753,
0.10379064083099365,
0.041681598871946335,
-0.1064034029841423,
0.05262702703475952,
-0.05310193449258804,
0.02190142311155796,
-0.04723012074828148,
0.04360583797097206,
-0.01125279814004898,
-0.04079202935099602,
-0.008189716376364231,
0.01068957895040512,
-0.0496620312333107,
0.010963698849081993,
-0.02294374257326126,
0.030469514429569244,
-0.05071762576699257,
-0.011168581433594227,
0.05147260054945946,
0.04645894840359688,
-0.04192475602030754,
-0.017734892666339874,
0.029539551585912704,
-0.028233714401721954,
0.06761913001537323,
0.029830919578671455,
-0.06630025058984756,
0.04527095705270767,
-0.01451683696359396,
-0.012205330654978752,
-0.009751495905220509,
-0.004101521801203489,
-0.011631585657596588,
-0.014242877252399921,
-0.09324363619089127,
-0.0858805850148201,
0.10758744180202484,
-0.06985289603471756,
0.040625687688589096,
0.01198164839297533,
-0.050047408789396286,
0.14338579773902893,
0.06342216581106186,
0.10180357843637466,
-0.01696300134062767,
-0.025686735287308693,
-0.010131427086889744,
-0.001195525866933167,
0.019773166626691818,
-0.046547774225473404,
-0.023219678550958633,
0.06080015003681183,
-0.06273284554481506,
0.00244241114705801
] |
49
Equations and inequalities
Example 9
Find the set of values of x for which:
3 β 5x β 2x2 < 0.
3 β 5 x β 2x2 = 0
2x2 + 5 x β 3 = 0
(2x β 1)( x + 3) = 0
x = 1 __ 2 or x = β 3
β3 1
2xy
O
So the required set of values is
x < β3 or x > 1 __ 2 .Multiply by β1 (so itβs easier to factorise).
1 _ 2 and β3 ar e the critical values.
Draw a sketch to show the shape of the graph
and the critical values.
Since the coefficient of x2 is negative, the graph
is βupside-down β-shapedβ. It crosses the x-axis at
β3 and 1 _ 2 . β Section 2.4
3 β 5x β 2x2 < 0 ( y < 0) for the outer parts of the
graph, belo
w the x-axis, as shown by the paler
parts of the curve.
In set notation this can be written as {x : x < β3} β {x : x >
1 _ 2 }.Quadratic equation.
Example 10
a Find the set of va lues of x for which 12 + 4x > x2.
b Hence find the set of va
lues for which 12 + 4x > x2 and 5x β 3 > 2.
a 12 + 4x > x2
0 > x2 β 4 x β 12
x2 β 4 x β 12 < 0
x2 β 4 x β 12 = 0
(x + 2)( x β 6) = 0
x = β2 or x = 6
Sketch of y = x2 β 4 x β 12
β26 xy
O
x2 β 4 x β 12 < 0
Solution: β 2 < x < 6You can use a table to check your solution.
β2 < x < 6
Use the critical values to split the real number
line into sets.
β26
x < β2 β2 < x < 6 x > 6
x + 2 β + +
x β 6 β β +
(x + 2)(x β 6) + β +
For each set, check whether the set of values makes the value of the bracket positive or negative. For example, if x < β2, (x + 2) is negative, (x β 6) is negative, and (x + 2)(x β 6) is (neg) Γ (neg) = positive.
In set notation the solution is {x : β2 < x < 6}.
|
[
0.05509074777364731,
0.05949096754193306,
0.08756668120622635,
-0.0159253291785717,
0.0013059937627986073,
-0.012959757819771767,
-0.03409389778971672,
0.0058800168335437775,
-0.12255316227674484,
0.05643250793218613,
-0.015162782743573189,
-0.09216229617595673,
-0.03740864247083664,
0.03606010600924492,
0.00830383412539959,
0.022363515570759773,
-0.022907067090272903,
-0.009563332423567772,
-0.052402909845113754,
-0.02561206929385662,
-0.002875734819099307,
-0.08516012132167816,
-0.08975107222795486,
-0.038288287818431854,
0.08030076324939728,
-0.05728466436266899,
-0.028104450553655624,
-0.06871286779642105,
0.014758133329451084,
-0.017224488779902458,
0.014054890722036362,
-0.039452847093343735,
0.057742372155189514,
-0.02030828967690468,
0.05445925146341324,
0.016755959019064903,
0.04949205368757248,
0.03138532489538193,
-0.029237430542707443,
-0.025136234238743782,
0.014000366441905499,
-0.019721759483218193,
-0.029841385781764984,
-0.023777587339282036,
0.03373783454298973,
0.0048505086451768875,
0.02326231822371483,
-0.047115739434957504,
0.003149240044876933,
0.023482494056224823,
0.005862475838512182,
0.04320235550403595,
0.05081181228160858,
0.08461765944957733,
-0.07044816762208939,
-0.1290685385465622,
0.034714847803115845,
-0.06228509917855263,
-0.11605538427829742,
-0.01260660495609045,
-0.01594569720327854,
-0.008366510272026062,
-0.048266056925058365,
0.059526391327381134,
-0.006780093070119619,
0.06141374632716179,
0.034759216010570526,
0.00006749672320438549,
-0.06851477175951004,
0.08037079125642776,
-0.14680172502994537,
0.02083793841302395,
0.009530534036457539,
-0.11835437268018723,
-0.033586956560611725,
0.0365789569914341,
-0.04514124616980553,
-0.06736398488283157,
-0.007467497605830431,
-0.01919212006032467,
0.015240970999002457,
0.021505914628505707,
0.04952148720622063,
0.012524789199233055,
-0.08680862188339233,
-0.0029883855022490025,
0.09286092966794968,
0.06545259058475494,
-0.017881788313388824,
0.029122887179255486,
0.05247678980231285,
0.029048558324575424,
0.028790928423404694,
0.013855303637683392,
0.06885286420583725,
-0.07257309556007385,
-0.030642924830317497,
-0.033173322677612305,
-0.02009725756943226,
0.049495600163936615,
0.009625950828194618,
-0.08863682299852371,
-0.004238460678607225,
0.027108244597911835,
-0.030478952452540398,
0.00014069373719394207,
0.0337371788918972,
-0.04422881826758385,
-0.014409467577934265,
-0.011630133725702763,
-0.02792329154908657,
-0.09423153102397919,
0.030498022213578224,
0.016837015748023987,
-0.009410763159394264,
0.008177840150892735,
0.020456330850720406,
0.09046587347984314,
0.020498165860772133,
-0.06520495563745499,
-0.005820857360959053,
-0.10109568387269974,
0.047752346843481064,
-0.04057225584983826,
-0.030595792457461357,
0.029658645391464233,
0.04192062467336655,
0.0384620800614357,
0.001167759415693581,
-0.088631771504879,
-0.06816641241312027,
-0.019782472401857376,
-0.004119011573493481,
-0.013739369809627533,
-0.020458586513996124,
-0.01866734027862549,
-0.049440063536167145,
0.07302086800336838,
0.002145481528714299,
0.016451910138130188,
-0.021692363545298576,
0.05289530009031296,
0.0550612173974514,
0.04123322665691376,
0.03844064474105835,
-0.013288931921124458,
-0.07198189944028854,
0.019276699051260948,
-0.06377145648002625,
-0.006838466972112656,
-0.0029250355437397957,
-0.018539492040872574,
0.0767766684293747,
0.1262955516576767,
-0.01898699626326561,
-0.12103062123060226,
-0.09654553979635239,
-0.03166135773062706,
0.005162321962416172,
0.016225358471274376,
-0.07844191789627075,
0.023740142583847046,
-0.007051846478134394,
0.11118268221616745,
-0.007799943909049034,
0.020602436736226082,
0.002197341062128544,
0.012333904393017292,
-0.030964786186814308,
-0.08204766362905502,
0.05773157626390457,
0.07481525093317032,
0.011337201111018658,
0.04784560948610306,
-0.0041797407902777195,
0.025221051648259163,
0.06329029053449631,
-0.0053505548276007175,
-0.031052859500050545,
0.09244830906391144,
0.03308381885290146,
-0.06346222013235092,
-0.02237805724143982,
-0.045773252844810486,
-0.009807289578020573,
-0.016197875142097473,
0.027444081380963326,
-0.028919769451022148,
0.014987883158028126,
-0.0021779679227620363,
0.03829989954829216,
-0.11759000271558762,
0.05797621235251427,
0.06168001890182495,
-0.09261073917150497,
-0.0724184438586235,
0.020079651847481728,
0.051059409976005554,
-0.09318773448467255,
-0.014625554904341698,
0.07997234910726547,
-0.026889534667134285,
0.14037595689296722,
0.05474013090133667,
-0.001090132282115519,
0.02082892134785652,
-0.05723854899406433,
-0.01848398521542549,
0.06625719368457794,
-0.0026353553403168917,
-0.056391630321741104,
0.05705677345395088,
-0.0715075209736824,
-0.008848868310451508,
-0.005527488421648741,
-0.057243797928094864,
0.051766231656074524,
-0.06946342438459396,
-0.012697204016149044,
0.025882398709654808,
-0.021828994154930115,
0.0016280364943668246,
0.06667739152908325,
-6.1497559186962196e-33,
-0.07897839695215225,
-0.024484878405928612,
-0.12236641347408295,
-0.06638367474079132,
0.0018066811608150601,
-0.008401638828217983,
0.020883265882730484,
-0.07621018588542938,
0.05447997897863388,
0.028535714372992516,
0.00013433804269880056,
0.028414329513907433,
0.005995430983603001,
-0.003055563196539879,
-0.02123616263270378,
-0.08478991687297821,
-0.04552751034498215,
0.041807472705841064,
0.03771335631608963,
-0.05639928579330444,
0.05969349294900894,
0.0002832179015967995,
-0.06323466449975967,
0.014249015599489212,
0.006765213795006275,
0.08622054755687714,
0.019092226400971413,
-0.06666240841150284,
-0.009935085661709309,
0.0028588564600795507,
-0.017021100968122482,
0.02512335777282715,
0.0654647946357727,
0.03748488798737526,
0.00025690990150906146,
-0.06009947508573532,
0.02493942156434059,
-0.05735080689191818,
0.09140357375144958,
-0.009133451618254185,
0.07003217190504074,
0.012573947198688984,
0.06628008186817169,
-0.014922389760613441,
-0.013831287622451782,
-0.0014590687351301312,
0.013616245239973068,
0.0652993842959404,
-0.015644434839487076,
0.034427400678396225,
0.0032252997625619173,
-0.06967375427484512,
-0.023646783083677292,
-0.0016490448033437133,
0.10635337233543396,
-0.030450692400336266,
0.03056287206709385,
0.022616561502218246,
0.00915464386343956,
0.03458067402243614,
-0.02410183846950531,
-0.07357075810432434,
0.04036489501595497,
0.014853973872959614,
-0.05900498479604721,
0.042607929557561874,
-0.03965320438146591,
-0.027473077178001404,
0.0006389031768776476,
-0.021428439766168594,
-0.039653848856687546,
-0.05997291952371597,
0.0022306530736386776,
-0.13338084518909454,
-0.08529890328645706,
0.045118048787117004,
-0.006615645717829466,
0.025194885209202766,
0.03711423650383949,
-0.12399479746818542,
-0.05807679519057274,
0.036271560937166214,
-0.013578413985669613,
0.04478360712528229,
-0.033989883959293365,
0.05461579188704491,
0.05966806784272194,
0.04728029668331146,
0.09197959303855896,
0.011781911365687847,
0.041349973529577255,
0.022767730057239532,
-0.005869044456630945,
-0.0393143966794014,
-0.007650941144675016,
6.358497560652952e-32,
-0.022360660135746002,
-0.0005996544496156275,
-0.03535725176334381,
0.052534069865942,
-0.02904500998556614,
-0.008908720687031746,
-0.06754040718078613,
-0.0423860140144825,
0.032213740050792694,
-0.03523816168308258,
0.12940549850463867,
0.10904891788959503,
-0.08999660611152649,
0.020832283422350883,
-0.02542552910745144,
0.03266063705086708,
-0.07651493698358536,
0.052404846996068954,
0.01277146115899086,
-0.019193509593605995,
0.01758461631834507,
0.0401080846786499,
-0.08311726152896881,
0.05619143694639206,
0.04856382682919502,
0.011359100230038166,
0.00010652803030097857,
-0.005849977023899555,
0.01583554595708847,
0.07028550654649734,
-0.0025998493656516075,
-0.011283345520496368,
-0.010094682686030865,
-0.056957196444272995,
0.1019418016076088,
0.061015743762254715,
-0.04436498507857323,
0.03938083350658417,
-0.040565501898527145,
-0.021478921175003052,
0.001913837157189846,
0.016111359000205994,
-0.007608127314597368,
-0.042546942830085754,
-0.004024284891784191,
-0.0673215314745903,
0.016407759860157967,
-0.07613997161388397,
0.05890943109989166,
-0.08170401304960251,
-0.05818301439285278,
0.15031971037387848,
0.04806965962052345,
0.07202521711587906,
0.005270824767649174,
-0.0025039007887244225,
-0.0025140326470136642,
0.03887717425823212,
-0.008949886076152325,
-0.02875882014632225,
0.03978215157985687,
0.08779483288526535,
-0.09681764245033264,
0.0101180924102664
] |
50
Chapter 3
Example 11
Find the set of values for which 6 __ x > 2 , x β 0b Solving 12 + 4 x > x2 gives β 2 < x < 6.
Solving 5 x β 3 > 2 gives x > 1.
β/four.ss01β202/four.ss0168
β2 < x < 6
x > 1
The two sets of values overlap where
1 < x < 6.
So the solution is 1 < x < 6.
6 __ x > x
6x
> 2x2
6x β 2x2 > 0
6x β 2x2 = 0
x(6 β 2 x) = 0
x = 0 or x = 3
3xy
O
The solution is 0 < x < 3.
1 Find the set of va lues of x for which:
a x2 β 11x + 24 < 0 b 12 β x β x2 > 0 c x2 β 3x β 10 > 0
d x2 + 7x + 12 > 0 e 7 + 13 x β 2x2 > 0 f 10 + x β 2x2 < 0
g 4x2 β 8x + 3 < 0 h β2 + 7x β 3x2 < 0 i x2 β 9 < 0
j 6x2 + 11x β 10 > 0 k x2 β 5x > 0 l 2x2 + 3x < 0
2 Find the set of va
lues of x for which:
a x2 < 10 β 3x b 11 < x2 + 10
c x(3 β
2x) > 1 d x(x
+ 11) < 3(1 β x2)Exercise 3EThis question is easier if you represent the
information in more than one way. Use a sketch graph to solve the quadratic inequality, and use a number line to combine it with the linear inequality.Problem-solving
In set notation this can be written as {x : 1 < x < 6}.
Solve the corresponding quadratic equation to find the critical values.
Sketch y = x (6 β 2 x). You are interested in the
values of x where the graph is above the x -axis.x = 0 can still be a critical value even though xΒ β Β 0. But it would not be part of the solution set, even if the inequality was > rather than > . x cou ld be either positive or negative,
so you canβt multiply both sides of this inequality by x . Instead, multiply both sides by x
2.
Because x2 is never negative, and x β 0 so x2 β 0,
the inequality sign stays the same.Watch out
|
[
0.024654731154441833,
0.07744352519512177,
0.008240856230258942,
0.018361078575253487,
-0.01585080660879612,
-0.015765078365802765,
-0.020603101700544357,
-0.022759582847356796,
-0.07385197281837463,
0.07337115705013275,
0.026658974587917328,
-0.02294725552201271,
0.030172277241945267,
-0.037580616772174835,
-0.019134286791086197,
0.008207175880670547,
-0.05009763687849045,
0.0020183860324323177,
-0.07257358729839325,
-0.01679474487900734,
0.008349139243364334,
-0.05170402675867081,
-0.04684269800782204,
0.012917543761432171,
0.08973636478185654,
-0.10401251912117004,
-0.009943741373717785,
-0.0588848702609539,
0.05416002869606018,
-0.001194394426420331,
0.02645273320376873,
-0.0018494396936148405,
0.02791614830493927,
-0.025971142575144768,
0.05603413283824921,
0.05869043990969658,
0.09402959793806076,
0.05992155522108078,
-0.08729491382837296,
0.03418093919754028,
-0.04893738403916359,
0.01422976516187191,
-0.02539031393826008,
-0.08103980869054794,
0.02132192812860012,
0.004081012215465307,
-0.03584500402212143,
0.006997228600084782,
-0.020169934257864952,
-0.043028879910707474,
0.023250054568052292,
-0.0013777677668258548,
0.0037439055740833282,
0.05847553163766861,
-0.11609841883182526,
-0.06521638482809067,
-0.028923075646162033,
-0.02898530662059784,
-0.08504420518875122,
0.0673980563879013,
-0.02955307997763157,
0.00915457308292389,
-0.011982939206063747,
0.0462590828537941,
0.03193502500653267,
0.05284459888935089,
0.0033590274397283792,
0.02055124193429947,
-0.024198226630687714,
0.05560678616166115,
-0.12816385924816132,
0.06382518261671066,
-0.0036711974535137415,
-0.01651979424059391,
0.019166836515069008,
0.020954271778464317,
-0.08053005486726761,
-0.05421802029013634,
-0.011176113970577717,
0.017977917566895485,
-0.08580343425273895,
-0.0021773295011371374,
0.07017861306667328,
0.0648384690284729,
-0.024204527959227562,
-0.005270111840218306,
0.10283831506967545,
-0.06385035067796707,
0.045158423483371735,
0.03543613478541374,
0.0839652568101883,
-0.0774109959602356,
-0.03834550082683563,
0.004964816849678755,
0.0051237489096820354,
-0.12840555608272552,
-0.007464257068932056,
-0.0012070754310116172,
0.025715602561831474,
0.08407842367887497,
0.03484481945633888,
-0.02108919993042946,
0.04292869567871094,
0.03676985576748848,
-0.01653897576034069,
-0.06506942212581635,
-0.040205758064985275,
-0.047849319875240326,
0.04199989512562752,
-0.11982547491788864,
0.010895686224102974,
-0.10060007870197296,
0.023639792576432228,
0.024866437539458275,
0.050291769206523895,
-0.039959266781806946,
0.0441964790225029,
0.004833265673369169,
-0.0036557933781296015,
-0.09006766229867935,
0.023203544318675995,
-0.06114009767770767,
-0.010352850891649723,
-0.03755221515893936,
-0.06417293846607208,
-0.027980227023363113,
0.07028509676456451,
0.01445694174617529,
-0.046242956072092056,
-0.10849485546350479,
-0.014963889494538307,
-0.07820690423250198,
-0.013837047852575779,
-0.005163040477782488,
0.005790464114397764,
-0.011694247834384441,
-0.05116232484579086,
0.10572758316993713,
0.018672913312911987,
-0.048350222408771515,
-0.020357467234134674,
0.020342541858553886,
0.008588630706071854,
-0.028435666114091873,
0.02856685034930706,
0.045649055391550064,
-0.06829442083835602,
-0.05102401226758957,
-0.0049684918485581875,
-0.023557137697935104,
-0.03279262036085129,
0.018857695162296295,
0.054523732513189316,
0.13577166199684143,
-0.028445033356547356,
-0.028614727780222893,
-0.030568165704607964,
0.005406874231994152,
0.011657025665044785,
0.046188730746507645,
-0.10999693721532822,
0.07485898584127426,
-0.014652185142040253,
0.0888073593378067,
-0.007611754350364208,
0.008444738574326038,
0.02624451369047165,
0.014405340887606144,
-0.04530033469200134,
-0.07963000237941742,
0.07662186026573181,
0.021721478551626205,
0.053808391094207764,
0.08066388219594955,
-0.055870648473501205,
0.07225370407104492,
0.033327121287584305,
0.013393607921898365,
0.004512233193963766,
0.05006597563624382,
0.01395330112427473,
-0.03403696417808533,
-0.054651159793138504,
-0.02647404931485653,
-0.020847363397479057,
0.016775459051132202,
-0.08753801882266998,
-0.07028668373823166,
0.0257699154317379,
-0.01178255770355463,
0.000786236603744328,
-0.0644986480474472,
0.03269191458821297,
0.0866081491112709,
-0.10226694494485855,
-0.0821210965514183,
0.02825414389371872,
0.04162894934415817,
-0.06409937888383865,
-0.030419986695051193,
0.013001547195017338,
-0.014379279688000679,
0.04771517589688301,
-0.020128043368458748,
0.016767732799053192,
-0.03744201362133026,
-0.03878175839781761,
-0.03446574509143829,
0.087870754301548,
-0.005194595083594322,
-0.023549003526568413,
-0.010493976064026356,
-0.07132986187934875,
-0.044185835868120193,
-0.01815742440521717,
-0.050293564796447754,
0.02166757360100746,
-0.06963784992694855,
-0.07200803607702255,
0.007503862492740154,
0.012261124327778816,
-0.022665465250611305,
0.06185334920883179,
-6.463305077782649e-33,
-0.08211511373519897,
0.02590678073465824,
-0.09945109486579895,
-0.05616052821278572,
0.026563173159956932,
-0.025490082800388336,
0.03171972930431366,
-0.010607894510030746,
0.022390568628907204,
-0.0008661982719786465,
0.013685494661331177,
0.007796014193445444,
-0.051426324993371964,
0.009422339498996735,
-0.041285838931798935,
-0.03662563115358353,
-0.01923743635416031,
-0.0005815971526317298,
0.048616837710142136,
0.05717941001057625,
0.06002978980541229,
0.05631769821047783,
-0.02742220088839531,
0.012246119789779186,
0.04386444017291069,
0.056554365903139114,
0.0946081131696701,
-0.0710836797952652,
-0.046107273548841476,
-0.023084836080670357,
-0.03459339588880539,
-0.009928666986525059,
0.13320468366146088,
-0.015170309692621231,
-0.035554006695747375,
-0.11160144954919815,
0.0592181533575058,
-0.031974148005247116,
-0.0012149635003879666,
-0.057988788932561874,
0.06059110909700394,
0.05086711049079895,
0.0009376141824759543,
-0.012919796630740166,
-0.02562153898179531,
0.06908509135246277,
0.06559254974126816,
0.08006109297275543,
0.027054594829678535,
0.022249994799494743,
-0.014424925670027733,
-0.026288222521543503,
0.05990457162261009,
0.04619865119457245,
0.08936372399330139,
-0.0101995337754488,
0.06529774516820908,
0.027507193386554718,
0.05150842294096947,
0.010172957554459572,
-0.017192145809531212,
-0.05468810722231865,
0.09852540493011475,
0.009627916850149632,
0.017582476139068604,
0.02922140620648861,
-0.05889921262860298,
-0.036256093531847,
0.013291374780237675,
0.03067065216600895,
-0.05635097995400429,
-0.008173682726919651,
-0.05808265507221222,
-0.13505591452121735,
0.0038895634934306145,
0.07914423942565918,
-0.016213055700063705,
0.03752991184592247,
0.03411859646439552,
-0.05798770859837532,
-0.1335795670747757,
0.030387436971068382,
0.02464226633310318,
-0.022370213642716408,
-0.11291151493787766,
0.050081316381692886,
0.0758724957704544,
0.05402457341551781,
0.07940027117729187,
-0.03135167807340622,
0.048535577952861786,
0.011301028542220592,
-0.022679435089230537,
-0.05698252469301224,
0.07749570161104202,
8.752481143673203e-32,
-0.003164837136864662,
-0.09343838691711426,
-0.07784375548362732,
0.015398875810205936,
0.045099757611751556,
0.05756299942731857,
-0.04955168813467026,
0.021518712863326073,
0.020360464230179787,
0.006539375986903906,
0.04233403503894806,
0.012414918281137943,
-0.023805828765034676,
-0.019931264221668243,
-0.06316465139389038,
0.0071868556551635265,
-0.023368997499346733,
0.032226406037807465,
0.00015412233187817037,
-0.0018821469275280833,
0.029765158891677856,
0.015814319252967834,
-0.05151461809873581,
0.03899950906634331,
0.051106054335832596,
0.016385072842240334,
-0.05425557494163513,
-0.00784339476376772,
0.027486946433782578,
-0.0014618762070313096,
0.04425601288676262,
-0.04835239797830582,
0.00045281529310159385,
0.02874099276959896,
0.09411393105983734,
0.0528840608894825,
-0.02448265813291073,
0.03147771209478378,
-0.09143194556236267,
-0.028398912400007248,
-0.02900419943034649,
-0.03911582753062248,
-0.08947566896677017,
0.019134970381855965,
0.030573852360248566,
-0.09404416382312775,
0.04729600250720978,
-0.02114187367260456,
0.08351294696331024,
-0.07472950220108032,
-0.05140705034136772,
0.1303156167268753,
0.02847500890493393,
0.07502679526805878,
0.0010252244537696242,
-0.06286456435918808,
-0.09262794256210327,
-0.005077693145722151,
-0.010186723433434963,
-0.058629877865314484,
0.0010338961146771908,
0.06534433364868164,
-0.08111057430505753,
0.020714789628982544
] |
51
Equations and inequalities
3 Use set notation to describe the set of v alues of x for which:
a x2 β 7x + 10 < 0 and 3x + 5 < 17 b x2 β x β 6 > 0 and 10 β 2x < 5
c 4x2 β 3x β 1 < 0 and 4(x + 2) < 15 β (x + 7) d 2x2 β x β 1 < 0 and 14 < 3x β 2
e x2 β x β 12 > 0 and 3x + 17 > 2 f x2 β 2x β 3 < 0 and x2 β 3x + 2 > 0
4 Given tha
t x β 0, find the set of values of x for which:
a 2 __ x < 1 b 5 > 4 __ x c 1 __ x + 3 > 2
d 6 + 5 __ x > 8 __ x e 25 > 1 ___ x 2 f 6 ___ x 2 + 7 __ x < 3
5 a Find the range of
values of k for which the
equation x2 β kx + (k + 3) = 0 has no real roots.
b Find the range of v
alues of p for which the
roots of the equation px2 + px β 2 = 0 are real.
6 Find the set of va
lues of x for which x2 β 5x β 14 > 0. (4 marks)
7 Find the set of va
lues of x for which
a 2(3x
β 1) < 4 β 3x (2 marks)
b 2x2 β 5x β 3 < 0 (4 marks)
c both 2(3x
β 1) < 4 β 3x and 2x2 β 5x β 3 < 0. (2 marks)
8 Given tha
t x β 3, find the set of values for which 5 _____ x β 3 < 2 .
(6 marks)
9 The equation kx2 β 2kx + 3 = 0, where k is a constant, has no real roots.
Prove that k satisfies the inequality 0 < k < 3. (4 marks)P
The quadratic equation ax2 + bx + c = 0
has real roots if b2 β 4ac > 0. β Section 2.5Hint
E
E
Multiply both sides of the
inequality by ( x β 3)2.Problem-solvingE/P
E/P
3.6 Inequalities on graphs
You may be asked to interpret graphically the solutions to inequalities by considering the graphs of
functions that are related to them.
β The values of x for which the curve y = f(x) is below the curve y = g(x) satisfy the inequality
f(x) < g( x).
β The values of x for which the curve y = f(x) is above the curve y = g(x) satisfy the inequality
f(x) > g( x).
|
[
0.02889794111251831,
0.09589443355798721,
0.09131527692079544,
-0.06901653856039047,
-0.005281904712319374,
0.07542191445827484,
-0.05173761025071144,
-0.008276140317320824,
-0.1181059256196022,
0.04395867511630058,
-0.04577496275305748,
-0.07411419600248337,
0.06039924919605255,
0.039319440722465515,
-0.0390905998647213,
0.005486749578267336,
0.0281234048306942,
-0.04202890396118164,
-0.03513072058558464,
-0.021378044039011,
0.019460860639810562,
-0.08012619614601135,
-0.06720908731222153,
-0.053684577345848083,
0.09319902211427689,
-0.05174282565712929,
-0.0413864441215992,
0.0020301134791225195,
-0.025695588439702988,
-0.019227970391511917,
0.009348166175186634,
-0.009669576771557331,
0.11505359411239624,
-0.03229226544499397,
0.035536594688892365,
0.0237965639680624,
0.061287421733140945,
0.023869358003139496,
-0.02918480895459652,
-0.0054885572753846645,
-0.03419658914208412,
-0.013524992391467094,
0.04598909243941307,
-0.03951852396130562,
-0.0019460677867755294,
0.03153460472822189,
0.033653806895017624,
-0.021455872803926468,
0.009546194225549698,
-0.011165698058903217,
0.034101393073797226,
0.007342393975704908,
0.011484871618449688,
0.14858613908290863,
-0.05641556531190872,
-0.11494322866201401,
0.0407380685210228,
-0.0604865737259388,
-0.03919867053627968,
-0.02079569362103939,
0.013228042051196098,
-0.04467133805155754,
0.01560561079531908,
0.015798956155776978,
0.03838647902011871,
0.04340715706348419,
0.014213562943041325,
0.036178868263959885,
-0.016052648425102234,
0.011839915998280048,
-0.13832414150238037,
0.07799425721168518,
0.011979823000729084,
-0.07390113919973373,
0.04640490934252739,
0.06993725150823593,
-0.027139905840158463,
-0.049457550048828125,
0.012827549129724503,
-0.0870773047208786,
-0.0033914269879460335,
-0.008694211952388287,
0.07901263982057571,
0.043815214186906815,
-0.025708038359880447,
-0.06421278417110443,
0.1243339255452156,
0.08472317457199097,
-0.015883684158325195,
-0.011550535447895527,
-0.010699514299631119,
-0.0015673977322876453,
0.031025191769003868,
-0.016839629039168358,
0.005383902695029974,
-0.1455935686826706,
0.026452502235770226,
-0.03429456800222397,
-0.019927484914660454,
0.07243888080120087,
0.030599897727370262,
0.004616953432559967,
0.03394537791609764,
0.010936640202999115,
-0.00300391367636621,
0.0058295088820159435,
0.0013493780279532075,
-0.04432349652051926,
-0.005559826735407114,
-0.08832216262817383,
-0.02639084681868553,
-0.08884534239768982,
0.03193996474146843,
0.06782137602567673,
-0.009433820843696594,
-0.07592076808214188,
0.0445224791765213,
0.022013381123542786,
-0.0036073026712983847,
-0.06770073622465134,
-0.052633874118328094,
-0.028201811015605927,
0.02817891538143158,
-0.015076818875968456,
-0.0020976366940885782,
-0.00820234976708889,
0.03298040106892586,
0.03776523470878601,
-0.093739815056324,
-0.1248311847448349,
-0.007511155214160681,
-0.02422751486301422,
0.03404344245791435,
-0.028087982907891273,
-0.0027471582870930433,
-0.06519247591495514,
-0.013704748824238777,
0.02318757027387619,
-0.06723281741142273,
-0.005038444884121418,
-0.014778828248381615,
0.08621285110712051,
0.07591038942337036,
0.015162901021540165,
0.07640805095434189,
-0.01716996729373932,
-0.06646884977817535,
0.013275341130793095,
-0.05455394461750984,
-0.013921424746513367,
-0.05261993035674095,
0.002654843032360077,
0.09353921562433243,
0.12411467730998993,
-0.09452646970748901,
-0.08129480481147766,
-0.06527259200811386,
-0.021435163915157318,
-0.0027872626669704914,
0.021307149901986122,
-0.042875051498413086,
0.0009076208807528019,
0.005679064430296421,
0.08333297818899155,
-0.011676663532853127,
0.026540610939264297,
0.011837652884423733,
0.013952251523733139,
-0.07338615506887436,
-0.03735547512769699,
0.039603590965270996,
0.04643779620528221,
-0.004194931592792273,
0.0540093369781971,
-0.01906757615506649,
-0.02817612700164318,
0.054934438318014145,
0.01773688569664955,
-0.09031462669372559,
0.0994790717959404,
0.04931254684925079,
-0.09170743823051453,
-0.028806161135435104,
-0.048603661358356476,
-0.07572285085916519,
0.024086294695734978,
-0.02950255200266838,
-0.017389584332704544,
-0.0010140122612938285,
-0.05573936179280281,
0.010394066572189331,
-0.0847863182425499,
0.041488949209451675,
0.058115404099226,
-0.06809363514184952,
-0.036793820559978485,
-0.0011801131768152118,
0.022539861500263214,
-0.09556221961975098,
0.011439184658229351,
0.027936343103647232,
0.004047109745442867,
0.053673408925533295,
-0.0015226893592625856,
-0.018437260761857033,
-0.012929869815707207,
-0.07493811100721359,
-0.045204658061265945,
0.057950906455516815,
0.037484295666217804,
-0.07198639959096909,
0.06244494020938873,
-0.07827454805374146,
-0.021898893639445305,
-0.028947891667485237,
-0.08004537224769592,
0.06508124619722366,
-0.10986915230751038,
-0.07198905944824219,
-0.024294478818774223,
-0.008710788562893867,
-0.027554579079151154,
0.04540088772773743,
-1.106442212703537e-32,
-0.058960795402526855,
0.009339656680822372,
-0.09739569574594498,
-0.07918746024370193,
0.0315910242497921,
-0.0009376159287057817,
0.06543669104576111,
-0.04001428559422493,
0.04715928062796593,
-0.0140775665640831,
0.09725288301706314,
-0.004091517999768257,
-0.037802956998348236,
-0.0368347093462944,
-0.05926290899515152,
-0.0412159264087677,
-0.05708467215299606,
0.01409906055778265,
0.040771398693323135,
0.021007340401411057,
0.01329873502254486,
-0.016256799921393394,
-0.04406765475869179,
-0.0036030984483659267,
-0.013981741853058338,
0.03176623582839966,
0.037882719188928604,
-0.014253932051360607,
-0.005243224091827869,
-0.04418354481458664,
-0.013033116236329079,
-0.03635106608271599,
0.060876619070768356,
0.07011266052722931,
-0.03674648329615593,
-0.05813553184270859,
0.02999858185648918,
-0.013943755999207497,
0.026489704847335815,
0.0011000391095876694,
0.05256464704871178,
0.042813338339328766,
0.047026168555021286,
0.06304012984037399,
0.0024190847761929035,
0.04078046604990959,
0.051202449947595596,
0.09231202304363251,
0.015611142851412296,
0.05096613988280296,
0.019672716036438942,
-0.03120991215109825,
-0.08427230268716812,
-0.0030077716801315546,
0.05863300710916519,
-0.013183845207095146,
-0.001120549044571817,
-0.01641612872481346,
0.024935944005846977,
-0.022450745105743408,
-0.02424224093556404,
-0.07676735520362854,
0.05119475722312927,
0.046703774482011795,
-0.057109370827674866,
0.07131777703762054,
-0.0588485524058342,
-0.041754450649023056,
-0.03462109714746475,
-0.04575594514608383,
0.03545575961470604,
-0.038041453808546066,
-0.010827284306287766,
-0.05377180129289627,
-0.0839708223938942,
-0.00007402887422358617,
0.012784093618392944,
0.09779846668243408,
0.0039856997318565845,
-0.012047204189002514,
-0.06146660074591637,
0.012162579223513603,
0.03525242954492569,
-0.01579231023788452,
-0.05630915239453316,
0.05311175808310509,
0.06488130241632462,
0.030680062249302864,
0.04782715067267418,
-0.02326957695186138,
0.059855952858924866,
0.03170301020145416,
0.015859637409448624,
-0.008659790270030499,
0.08371415734291077,
8.589090367644782e-32,
-0.006501685827970505,
-0.013123449869453907,
-0.07548975199460983,
0.015975292772054672,
0.03855002298951149,
-0.022351577877998352,
-0.024609746411442757,
0.00473271356895566,
-0.03908083215355873,
-0.008434196002781391,
0.1281040608882904,
0.018291015177965164,
-0.11712466180324554,
0.0023591758217662573,
0.00029773396090604365,
-0.02610028348863125,
-0.06443589180707932,
0.03566333279013634,
0.03337942808866501,
-0.0355236791074276,
-0.017554011195898056,
0.04838509112596512,
-0.04084128513932228,
0.048223745077848434,
0.030022086575627327,
-0.009962116368114948,
-0.06323754042387009,
0.005462624132633209,
0.05642498657107353,
0.037662554532289505,
0.050826556980609894,
0.007448410149663687,
-0.004940354265272617,
-0.09491779655218124,
0.10535135120153427,
0.07216812670230865,
-0.03529234975576401,
-0.0077211675234138966,
-0.012649043463170528,
-0.040097132325172424,
0.06357134133577347,
0.0053945258259773254,
-0.023252593353390694,
-0.06346268206834793,
0.0104092787951231,
-0.07937207818031311,
0.019256645813584328,
-0.029309861361980438,
0.03489316627383232,
-0.1040983721613884,
-0.05117829889059067,
0.1348109096288681,
0.015692779794335365,
0.04439661279320717,
-0.01937701180577278,
0.02905457839369774,
-0.03109678067266941,
0.04139496758580208,
0.0036758023779839277,
-0.12040553241968155,
0.024863174185156822,
0.12811149656772614,
-0.06111537292599678,
0.015941018238663673
] |
52
Chapter 3
Example 12
L1 has equation y = 12 + 4x.
L2 has equation y = x2.
The diagram shows a sketch of L1 and L2 on the same axes.
a Find the coordinates of
P1 and P2, the points of intersection.
b Hence write down the solution to the inequality 12
+ 4x > x2.y
x OL1: y = 12 + 4x
L2: y = x2P1
P2
a x2 = 12 + 4 x
x2 β 4 x β 12 = 0
(x
β 6)( x + 2) = 0
x = 6 and x = β 2
substitute into y = x2
when x = 6, y = 36 P1 (6, 36)
when x = β 2, y = 4 P2 (β2, 4)
b 12 +
4x > x2 when the graph of L1 is
above the graph of L2
β2 < x < 6This is the range of values of x for which the
graph of y = 12 + 4x is above the graph of y = x2
i.e. between the two points of intersection.
In set notation this is {x : β2 < x < 6}.Equate to find the points of intersection, then
rearrange to solve the quadratic equation.
Factorise to find the x-coordinates at the points of intersection.
1 L1 has equation 2y + 3x = 6.
L2 has the equation x β y = 5.
The diagram shows a sketch of L1 and L2.
a Find the coordinates of
P, the point of intersection.
b Hence write down the solution to the inequality
2
y + 3x > x β y.y
x O
L1: 2y + 3x = 6L2: x β y = 5Exercise 3Fy
x 2 5 Oy = g(x)y = f(x)
The solutions to f(x) = g(x) are x = 2 and x = 5.f(x) is below g(x) when 2 < x < 5. These values of
x satisfy f(x) < g(x).f(x) is above g(x) when x < 2 and when x > 5.
These values of x satisfy f(x) > g(x).
|
[
0.07161044329404831,
0.05860932171344757,
-0.00026715637068264186,
-0.02531319297850132,
0.005367380101233721,
-0.025871116667985916,
-0.0031512973364442587,
0.0499584823846817,
-0.09999948740005493,
0.044180333614349365,
0.03993464633822441,
-0.0017477524233981967,
0.0232989601790905,
0.037575483322143555,
-0.07471968233585358,
-0.02958429418504238,
-0.014100436121225357,
0.0003206884430255741,
-0.05549636483192444,
-0.03480372205376625,
-0.025886915624141693,
-0.11209581047296524,
-0.04915647953748703,
-0.04596838727593422,
0.05085631087422371,
-0.13798320293426514,
-0.054638899862766266,
-0.04235769808292389,
0.0368817001581192,
0.03189685568213463,
0.10303085297346115,
0.006604885682463646,
0.024178754538297653,
0.025883778929710388,
0.01312243938446045,
0.04401649162173271,
0.08943764120340347,
0.01485997624695301,
0.03196591138839722,
-0.05764235556125641,
-0.09374872595071793,
-0.07027173787355423,
-0.004978940822184086,
-0.0986897274851799,
0.02256501279771328,
-0.07456636428833008,
-0.04087371006608009,
0.022842759266495705,
0.03302445262670517,
-0.05193017050623894,
0.04641206935048103,
-0.03430239111185074,
0.009151151403784752,
0.02719142846763134,
-0.01939699612557888,
0.04179299250245094,
-0.015436193905770779,
-0.029163794592022896,
0.03153175115585327,
0.09467294067144394,
0.07896580547094345,
-0.0587121844291687,
-0.054389920085668564,
0.021597348153591156,
0.03283533826470375,
0.008910676464438438,
-0.0022095271851867437,
0.04921962320804596,
-0.06616440415382385,
0.08468450605869293,
-0.12984894216060638,
0.03934943303465843,
-0.004971430636942387,
-0.07616209983825684,
0.02077750489115715,
-0.0386338047683239,
-0.030137378722429276,
-0.06619628518819809,
-0.03004549816250801,
-0.04773455485701561,
-0.0365755520761013,
0.06468871235847473,
-0.03175235912203789,
0.024719655513763428,
-0.031250499188899994,
0.0010215917136520147,
0.06088510900735855,
-0.06002327799797058,
0.07343194633722305,
0.019274329766631126,
0.03453497588634491,
0.020824555307626724,
-0.02941671945154667,
-0.009576904587447643,
0.030136484652757645,
-0.16633310914039612,
0.07990077883005142,
0.031582824885845184,
0.03910175710916519,
0.1085711270570755,
-0.012161729857325554,
0.016497377306222916,
-0.012186786159873009,
0.03953798860311508,
0.053734853863716125,
0.021891942247748375,
0.010818527080118656,
-0.032891470938920975,
0.02754276990890503,
-0.06393172591924667,
0.013156168162822723,
-0.05129040405154228,
0.0026248274371027946,
0.06832051277160645,
0.09398883581161499,
-0.012977711856365204,
0.06526932865381241,
-0.048735007643699646,
-0.015757806599140167,
-0.04328390210866928,
0.01755775883793831,
-0.04024170711636543,
-0.03406064212322235,
-0.03827786073088646,
-0.014850799925625324,
-0.04001644253730774,
-0.00806387234479189,
0.004759570583701134,
-0.0007152666221372783,
-0.06498990207910538,
-0.03350922837853432,
-0.04326673224568367,
0.008278338238596916,
-0.08188779652118683,
-0.0025597757194191217,
-0.01943269744515419,
-0.028292030096054077,
0.08238831907510757,
-0.046197470277547836,
-0.058258116245269775,
0.060058336704969406,
-0.013916374184191227,
0.05378280207514763,
0.015734579414129257,
-0.011206637136638165,
0.023168770596385002,
-0.09638551622629166,
-0.012181051075458527,
0.0225787665694952,
0.00044697950943373144,
-0.012789178639650345,
-0.0287412591278553,
-0.01912645250558853,
0.13176079094409943,
-0.02139163948595524,
-0.030364319682121277,
-0.030142806470394135,
-0.013471231795847416,
0.03870675712823868,
0.02715030126273632,
-0.06384588778018951,
0.047832489013671875,
-0.004471576306968927,
0.026466453447937965,
-0.03040069341659546,
-0.006429403554648161,
-0.02275335043668747,
0.11969725787639618,
0.00974200014024973,
-0.05579695850610733,
0.04356474429368973,
0.031254660338163376,
0.0002500558039173484,
0.04332791268825531,
-0.06264057010412216,
-0.03336277976632118,
-0.0027234936133027077,
-0.039919592440128326,
-0.027976779267191887,
0.07447680085897446,
-0.01213799323886633,
-0.11049801856279373,
0.03739887848496437,
-0.008758582174777985,
-0.004196662921458483,
0.02997792512178421,
0.04899787902832031,
0.006090361159294844,
-0.06688253581523895,
0.04532988369464874,
0.023702947422862053,
-0.08003934472799301,
-0.01927373930811882,
0.08162697404623032,
-0.10120438039302826,
-0.09276144951581955,
0.06913488358259201,
0.04197799786925316,
-0.05383063480257988,
-0.03537924587726593,
-0.035242531448602676,
0.035195086151361465,
0.01071816124022007,
-0.08023620396852493,
-0.05777248740196228,
0.038344964385032654,
-0.01562570594251156,
-0.0029320635367184877,
0.003597768722102046,
0.01734156161546707,
-0.0027360988315194845,
-0.017256734892725945,
-0.08682763576507568,
-0.029950831085443497,
0.00020170699281152338,
0.0036036204546689987,
0.09622278064489365,
-0.01890280283987522,
-0.1047964096069336,
-0.008911654353141785,
-0.028247641399502754,
0.020397638902068138,
0.10860128700733185,
-6.085121729158241e-33,
-0.011247389949858189,
0.06826823204755783,
-0.048964839428663254,
-0.1165633350610733,
-0.025466421619057655,
0.005835835821926594,
0.11524965614080429,
-0.06009555980563164,
0.05978509783744812,
0.08184126764535904,
0.0579042062163353,
-0.010905128903687,
-0.025348959490656853,
0.013518223538994789,
-0.006171848624944687,
-0.052990466356277466,
-0.008010423742234707,
-0.014658895321190357,
-0.027364581823349,
-0.027851171791553497,
0.006584505550563335,
-0.0043959226459264755,
-0.020719759166240692,
-0.06144427880644798,
0.011460965499281883,
0.01323486678302288,
0.07164411246776581,
-0.03547794371843338,
-0.07188691198825836,
0.008581879548728466,
0.008773455396294594,
-0.04937474802136421,
0.0697360411286354,
0.021941479295492172,
0.0057077184319496155,
-0.04380728676915169,
-0.02381446212530136,
-0.04071289300918579,
0.06261855363845825,
-0.11575523763895035,
0.04203192517161369,
0.04114176705479622,
0.026776116341352463,
-0.004361483734101057,
0.005211992189288139,
0.05433616042137146,
0.08151083439588547,
0.0775497779250145,
0.006040878128260374,
0.010579337365925312,
0.018610401079058647,
-0.006986734922975302,
0.033043332397937775,
-0.0785130113363266,
0.10338642448186874,
-0.021073533222079277,
0.023487884551286697,
-0.045018505305051804,
0.009217418730258942,
-0.04204277694225311,
0.010633138939738274,
0.003972355742007494,
0.05682555213570595,
0.04306420683860779,
-0.0059293108060956,
0.0005370483850128949,
-0.05642620101571083,
-0.050580866634845734,
0.09195239841938019,
0.0026913403999060392,
-0.08332037180662155,
-0.01506266463547945,
-0.031033135950565338,
-0.0677371695637703,
0.03291219845414162,
0.038410402834415436,
-0.012355905957520008,
0.04129777103662491,
0.07809358835220337,
-0.03826909512281418,
-0.08583259582519531,
0.055489689111709595,
-0.030109120532870293,
-0.03701147809624672,
-0.06374193727970123,
0.08469022065401077,
-0.009109719656407833,
0.044860731810331345,
0.029864918440580368,
-0.018741659820079803,
0.029011311009526253,
0.02528817020356655,
-0.04608943313360214,
-0.02532108686864376,
0.08046787232160568,
8.459030733933927e-32,
-0.0011285695945844054,
0.01701929047703743,
-0.05753788724541664,
-0.0563737116754055,
0.04975474625825882,
0.09562062472105026,
0.058715399354696274,
0.0900610089302063,
0.046871062368154526,
0.01782754622399807,
0.007756057195365429,
-0.0056781102903187275,
0.01173649076372385,
-0.025835441425442696,
-0.06527600437402725,
0.0056664771400392056,
-0.026563145220279694,
-0.050373539328575134,
0.026178212836384773,
-0.07160693407058716,
-0.06167948991060257,
-0.0396418534219265,
-0.02263479307293892,
0.051990289241075516,
0.035071175545454025,
0.06473556160926819,
-0.08579950779676437,
0.00370183028280735,
0.05658606067299843,
-0.07789543271064758,
0.07706569880247116,
-0.07786767184734344,
0.03232776001095772,
0.0024596417788416147,
0.12315703928470612,
-0.05072987824678421,
-0.07832976430654526,
0.03976494446396828,
0.020442843437194824,
-0.038463495671749115,
-0.02823886089026928,
-0.047483522444963455,
-0.050822388380765915,
-0.035818032920360565,
0.04225750267505646,
0.04292803630232811,
0.08147158473730087,
-0.07398810982704163,
0.08140217512845993,
-0.029796214774250984,
-0.0586579293012619,
0.08069286495447159,
0.03471499681472778,
0.06132170557975769,
-0.03243463858962059,
-0.09818579256534576,
0.002940710633993149,
0.019940733909606934,
-0.0065459078177809715,
-0.020109685137867928,
-0.028918173164129257,
0.04678616672754288,
-0.11194227635860443,
-0.03327533230185509
] |
53
Equations and inequalities
The sketch shows the graphs of
f(x) = x2 β 4x β 12
g(x) = 6 + 5 x β x2
a Find the coordinates of the points of intersection.
b Fin
d the set of values of x for which f( x) < g( x).
Give your answer in set notation.y
x O
y = g(x)y = f (x)Challenge
All the shaded points in this region satisfy the
inequality y > f(x).3.7 Regions
You can use shading on graphs to identify regions that satisfy linear and quadratic inequalities.
β y <
f(x) represents the points on the coordinate grid below the curve y = f(x).
β y >
f(x) represents the points on the coordinate grid above the curve y = f(x).
y = f (x)y
x OAll the unshaded points in this region satisfy the
inequality y < f(x).2 For each pair of functions:
i Sketch the gra
phs of y = f(x) and y = g(x) on the same axes.
ii Find the coordinates of
any points of intersection.
iii Write down the solutions to the inequa
lity f(x) < g(x).
a f(x
) = 3x β 7 b f(x
) = 8 β 5x c f(x
) = x2 + 5
g(x
) = 13 β 2x g(x
) = 14 β 3x g(x)
= 5 β 2x
d f(x
) = 3 β x2 e f(x ) = x2 β 5 f f(x ) = 7 β x2
g(x
) = 2x β 12 g(x)
= 7x + 13 g(x)
= 2x β 8
3 Find the set of va
lues of x for which the curve with equation y = f(x) is below the line with
equation y = g(x).a
f(x
) = 3x2 β 2x β 1 b f(x ) = 2x2 β 4x + 1 c f(x ) = 5x β 2x2 β 4
g(x
) = x + 5 g(x)
= 3x β 2 g(x)
= β2x β 1
d f(x
) = 2 __ x , x β 0 e f(x) = 3 __ x2 β 4 __ x , x β 0 f f(x) = 2 _____ x + 1 , x β β1
g(x
) = 1 g(x)
= β1 g(x)
= 8P
|
[
0.0902569517493248,
0.07046928256750107,
0.07022906839847565,
-0.032154425978660583,
-0.04011542722582817,
0.025275414809584618,
0.026603639125823975,
0.004776309709995985,
-0.15033164620399475,
0.04792482778429985,
-0.04281258583068848,
-0.026247045025229454,
-0.040408939123153687,
0.02789643220603466,
-0.07477625459432602,
-0.02739955484867096,
-0.0008401369559578598,
0.004337845835834742,
-0.0766797885298729,
-0.07660830765962601,
-0.007792371790856123,
-0.095893993973732,
-0.01964309625327587,
-0.06531558185815811,
0.07063283771276474,
-0.13435155153274536,
-0.058887165039777756,
-0.06014736369252205,
-0.0045826924033463,
0.0026177780237048864,
0.08289150148630142,
-0.04258264601230621,
0.01392043475061655,
-0.04801906272768974,
0.013844645582139492,
0.04332651570439339,
0.014755324460566044,
0.04732964560389519,
0.06100078299641609,
-0.013103032484650612,
-0.09466487169265747,
-0.03566145524382591,
-0.02051069214940071,
-0.025544101372361183,
0.061023253947496414,
-0.002651025541126728,
0.02796601504087448,
0.006057672202587128,
-0.025046689435839653,
0.01630990393459797,
0.022006582468748093,
-0.026260575279593468,
-0.030968770384788513,
0.08160790055990219,
-0.028476262465119362,
-0.045117732137441635,
0.05126672983169556,
-0.08495599031448364,
0.009717357344925404,
0.09763876348733902,
0.02965371496975422,
0.02866748906672001,
-0.04431098699569702,
0.04966725409030914,
0.04094879329204559,
0.06310475617647171,
0.07606253772974014,
-0.004787040874361992,
-0.047117576003074646,
0.07915998995304108,
-0.11408828943967819,
0.036631178110837936,
-0.03641390800476074,
-0.08042693883180618,
-0.005442953202873468,
0.02308015152812004,
-0.05240378901362419,
-0.062213826924562454,
-0.013824068009853363,
-0.05288085713982582,
-0.04312725365161896,
0.05516976863145828,
0.032885562628507614,
0.10311771929264069,
-0.014352815225720406,
0.008988719433546066,
0.04322274401783943,
0.0019801852758973837,
0.005338555201888084,
0.04418858140707016,
-0.007086507510393858,
0.02603628858923912,
-0.007961235009133816,
-0.061839986592531204,
-0.0032425473909825087,
-0.13503903150558472,
0.08018404245376587,
-0.0009664234239608049,
0.0022542220540344715,
0.11467932909727097,
0.016262950375676155,
0.02697344869375229,
0.03701576963067055,
0.013119969516992569,
0.028897838667035103,
-0.013583636842668056,
-0.028066936880350113,
-0.0035061780363321304,
-0.028485119342803955,
-0.033895429223775864,
-0.023937487974762917,
-0.02935369312763214,
0.010926634073257446,
0.050265196710824966,
0.04914040490984917,
-0.05921809375286102,
0.07652512192726135,
-0.04152149707078934,
0.02470516227185726,
-0.04253528267145157,
-0.007150681689381599,
-0.050802603363990784,
0.020309044048190117,
-0.0005486853187903762,
-0.02356073074042797,
-0.04724327102303505,
-0.00044163077836856246,
0.011121965944766998,
0.019450530409812927,
-0.10425326228141785,
0.023712029680609703,
-0.011555965058505535,
0.04992980137467384,
-0.05400371551513672,
-0.009091474115848541,
-0.04761861264705658,
-0.04695506393909454,
0.08984459936618805,
-0.00975114107131958,
-0.007249580696225166,
-0.009340928867459297,
0.08733364194631577,
0.11996787786483765,
0.028458600863814354,
0.0075554740615189075,
0.014943123795092106,
-0.09946514666080475,
-0.00317612336948514,
-0.006170892622321844,
-0.042084407061338425,
0.0239635668694973,
-0.003300188574939966,
0.02434365637600422,
0.19250400364398956,
-0.01755368523299694,
-0.04138069227337837,
-0.08236788958311081,
0.017049845308065414,
-0.009070741012692451,
-0.027456820011138916,
-0.05829013139009476,
0.09721755236387253,
-0.016604170203208923,
0.04289139807224274,
-0.048704277724027634,
-0.0051392377354204655,
0.01979217678308487,
0.02719922736287117,
-0.04062006250023842,
-0.056420180946588516,
0.05078703910112381,
0.018989019095897675,
0.002692939480766654,
0.054377004504203796,
-0.02080393023788929,
-0.06754234433174133,
0.025461705401539803,
0.006696593482047319,
-0.06773748993873596,
0.0709322914481163,
0.025279516354203224,
-0.07756690680980682,
0.030076637864112854,
-0.03808116912841797,
0.011311696842312813,
-0.03479647636413574,
0.011441316455602646,
-0.040460266172885895,
-0.04642655700445175,
-0.011820525862276554,
0.006643247324973345,
-0.08976218104362488,
0.0011353917652741075,
0.020446542650461197,
-0.11209224164485931,
-0.040799856185913086,
-0.0023565373849123716,
0.06263529509305954,
-0.07381041347980499,
-0.01834619604051113,
0.05528445169329643,
-0.002861263230443001,
0.0463654026389122,
-0.02613358199596405,
-0.004381421487778425,
-0.015937164425849915,
-0.06861045956611633,
-0.013237244449555874,
0.04735655337572098,
0.0522233247756958,
-0.05381402000784874,
-0.0003029352519661188,
-0.1136927381157875,
0.031073983758687973,
0.0929921567440033,
-0.05473766475915909,
0.10157115012407303,
-0.0527355931699276,
-0.1265369951725006,
-0.04240136593580246,
-0.014453006908297539,
-0.036119069904088974,
0.06029316782951355,
1.732396921809839e-33,
-0.02821439318358898,
0.011346726678311825,
-0.06800787150859833,
-0.06039759889245033,
0.019755449146032333,
-0.04920225962996483,
0.09142867475748062,
-0.05019082501530647,
0.060383375734090805,
0.054976481944322586,
0.07051532715559006,
0.022806640714406967,
0.015264298766851425,
0.025898147374391556,
-0.033630989491939545,
-0.05778946354985237,
-0.03409336134791374,
-0.0199953094124794,
-0.03750922530889511,
0.00562116876244545,
0.038043063133955,
-0.00252217473462224,
-0.003324701450765133,
-0.027194714173674583,
0.04267718642950058,
0.023755360394716263,
0.08493531495332718,
-0.05507250130176544,
-0.06065697595477104,
0.05809119716286659,
0.026314781978726387,
0.004874954465776682,
0.05335061252117157,
0.0268096923828125,
0.03313539922237396,
-0.09464099258184433,
0.0541657879948616,
-0.04753193259239197,
0.08904975652694702,
-0.050471965223550797,
0.052981290966272354,
0.061859190464019775,
0.07249189913272858,
0.028728947043418884,
-0.01771824061870575,
0.0457332469522953,
-0.013264305889606476,
0.04069714620709419,
-0.03194029629230499,
0.03850915655493736,
0.008280240930616856,
-0.05527694523334503,
0.019565746188163757,
-0.013925083912909031,
0.12007810175418854,
-0.005376155953854322,
0.066468246281147,
-0.04476028308272362,
0.020517902448773384,
0.054571762681007385,
-0.03521731123328209,
-0.1009640246629715,
0.05656322091817856,
-0.006571678444743156,
-0.05204673483967781,
-0.03672162815928459,
-0.10000384598970413,
-0.06964776664972305,
0.025758890435099602,
0.023872138932347298,
-0.04003558307886124,
-0.043543532490730286,
-0.026671644300222397,
-0.06770839542150497,
0.023097719997167587,
0.042014844715595245,
-0.0005291630513966084,
0.0248397346585989,
0.02525367960333824,
-0.020494556054472923,
-0.048814963549375534,
0.05753589794039726,
0.018762847408652306,
0.025236545130610466,
-0.07048066705465317,
0.1017310842871666,
0.02484414167702198,
0.03159995749592781,
0.07988761365413666,
-0.010130529291927814,
0.0006880550063215196,
0.1088332086801529,
-0.08999460190534592,
-0.03294185921549797,
0.030045736581087112,
6.374712329728195e-32,
-0.042139723896980286,
0.04082421585917473,
-0.02825061045587063,
0.03168441727757454,
0.0552683025598526,
0.03668951988220215,
0.008647005073726177,
0.014989008195698261,
0.06353418529033661,
-0.037243131548166275,
0.07486465573310852,
-0.008633394725620747,
-0.11214791983366013,
-0.02945476956665516,
0.0026783510111272335,
0.01676206849515438,
-0.053169529885053635,
0.01632518880069256,
0.007062061689794064,
-0.018361613154411316,
-0.02845144271850586,
0.014733238145709038,
-0.05580732598900795,
0.06560945510864258,
0.03376483544707298,
0.0281230416148901,
-0.04726870730519295,
-0.02285717986524105,
-0.005092869512736797,
0.02869861200451851,
-0.009055979549884796,
-0.0359044075012207,
-0.03945288807153702,
0.004080704879015684,
0.1169201135635376,
0.020233988761901855,
-0.10700328648090363,
0.02126404084265232,
-0.025879835709929466,
-0.032454632222652435,
-0.007863185368478298,
0.0036488594487309456,
0.034696314483881,
-0.016801228746771812,
-0.012897799722850323,
-0.06269999593496323,
0.0850939080119133,
-0.006437013857066631,
0.05241818353533745,
0.0065301936119794846,
-0.06060526892542839,
0.10496965050697327,
0.002548074582591653,
0.0004332885146141052,
-0.05657432600855827,
0.0031860885210335255,
0.014757905155420303,
0.014142082072794437,
-0.02322630025446415,
-0.03271862864494324,
-0.030324788764119148,
0.06585965305566788,
-0.10930556803941727,
0.036536335945129395
] |
54
Chapter 3
β If y
> f(x) or y < f(x) then the curve y = f(x) is not included in the region and is represented
by a dotted line.
β If y > f(x) or y < f(x) then the curve y = f(x) is included in the region and is represented by a
solid line.
Example 13
On graph paper, shade the region that satisfies the inequalities:
y > β2, x < 5, y < 3x + 2 and x > 0.
y
x
β5β2 β1 1234 567 β35
O10152025x = 0 x = 5
y = 3x + 2
y = β2Draw dotted lines for x = 0, x = 5.
Draw solid lines for y = β2, y = 3x + 2.Shade the required region.
Test a point in the region. Try (1, 2).For x = 1: 1 < 5 and 1 > 0 βFor y = 2: 2 > β2 and 2 < 3 + 2 β
Example 14
On graph paper, shade the region that satisfies the inequalities:
2y + x < 14
y > x2 β 3x β 4
y
x
β5β1 12 3/four.ss01 567 β2 β3 β/four.ss015
O1015202530
β10y = x2 β 3x β /four.ss01
2y + x = 1/four.ss01Draw a dotted line for 2y + x = 14 and a
solid line for y = x2 β 3x β 4.
Shade the required region.Test a point in the region. Try (0, 0). 0 + 0 < 14 and 0 > 0 β 0 β 4 β Explore which regions on
the g
raph satisfy which inequalities
using GeoGebra.Online
|
[
0.031383588910102844,
0.08265526592731476,
0.021256988868117332,
-0.05200735852122307,
-0.022602982819080353,
-0.010243067517876625,
-0.09278827160596848,
0.00941680558025837,
-0.09922920912504196,
0.03958097845315933,
0.0057642157189548016,
0.03810950741171837,
0.0738588273525238,
0.08055519312620163,
-0.058318961411714554,
-0.08605489879846573,
-0.04445939511060715,
0.006835577078163624,
-0.0505237802863121,
-0.04859897121787071,
-0.0008101736893877387,
-0.05031643807888031,
-0.10034731775522232,
-0.025079069659113884,
0.03371330723166466,
-0.10612897574901581,
-0.05846892669796944,
-0.03003920242190361,
-0.010867646895349026,
-0.012374822981655598,
0.09765685349702835,
-0.03232144936919212,
-0.011443433351814747,
-0.03003171645104885,
0.024712443351745605,
-0.044647183269262314,
0.04001889377832413,
-0.0017287948867306113,
0.08537034690380096,
-0.012449764646589756,
-0.08565653860569,
0.010212216526269913,
-0.01739593595266342,
0.0755690410733223,
0.048210788518190384,
-0.010446769185364246,
0.030733522027730942,
-0.0829119086265564,
0.021071970462799072,
-0.000669775006826967,
0.06972512602806091,
0.05466637760400772,
-0.014108303934335709,
0.030884472653269768,
-0.045170534402132034,
-0.0036312376614660025,
0.010565576143562794,
-0.03375281020998955,
-0.10242252051830292,
0.14097489416599274,
0.024205295369029045,
0.06190309301018715,
-0.005417650565505028,
0.03741440922021866,
0.09033797681331635,
0.040241654962301254,
0.032265402376651764,
-0.026891322806477547,
-0.05342099443078041,
0.1308392733335495,
-0.1347751021385193,
-0.06976626068353653,
-0.026545299217104912,
-0.07885053753852844,
-0.010957556776702404,
-0.003944744355976582,
-0.0219595767557621,
0.030034426599740982,
-0.03481269255280495,
-0.05052261799573898,
-0.05226275697350502,
0.07300914824008942,
0.02635275386273861,
0.10205188393592834,
-0.049467846751213074,
0.040257468819618225,
0.030244629830121994,
-0.05471443757414818,
-0.008812193758785725,
0.08543293923139572,
-0.04017845168709755,
0.06245295703411102,
-0.0384662039577961,
-0.030707266181707382,
0.012385048903524876,
-0.058441754430532455,
-0.0007234481745399535,
-0.032392051070928574,
-0.004499183502048254,
0.05425553396344185,
-0.006349653005599976,
-0.09236271679401398,
-0.005420372821390629,
0.005946133751422167,
0.02557157352566719,
-0.06370456516742706,
0.042357608675956726,
0.002724012127146125,
-0.04629432410001755,
-0.0332103855907917,
0.004527426324784756,
0.005070623010396957,
0.000044005686504533514,
0.054101377725601196,
0.06964104622602463,
-0.12097135931253433,
-0.03353624418377876,
0.008317719213664532,
0.030471784994006157,
-0.06682990491390228,
-0.0046574268490076065,
-0.019035672768950462,
-0.04861126467585564,
-0.016385354101657867,
-0.013189682736992836,
-0.04035300761461258,
0.006602390669286251,
-0.030664851889014244,
0.04326067492365837,
-0.10176648944616318,
0.004548171069473028,
-0.05800490453839302,
0.06897015124559402,
-0.05583149939775467,
0.0027446961030364037,
-0.0646917074918747,
-0.08980796486139297,
0.07571379840373993,
0.007076576352119446,
0.0031159210484474897,
0.025961099192500114,
0.02530374750494957,
0.1055753231048584,
0.016309423372149467,
0.020588237792253494,
-0.004545563366264105,
-0.05185738205909729,
0.02275528386235237,
0.026209034025669098,
-0.06688585877418518,
0.013176762498915195,
-0.013521615415811539,
0.019896456971764565,
0.10198651254177094,
0.005445555318146944,
-0.00786782056093216,
-0.08153475075960159,
0.04313056915998459,
0.017963185906410217,
-0.04125748202204704,
-0.041792403906583786,
0.09401071816682816,
-0.031157374382019043,
0.051842041313648224,
-0.1043282002210617,
-0.015488623641431332,
0.05471387878060341,
-0.012534596025943756,
-0.004532505292445421,
-0.03934058919548988,
0.054830990731716156,
-0.005933630745857954,
-0.029685908928513527,
0.07864369451999664,
-0.018553948029875755,
0.03721761703491211,
0.022408541291952133,
0.011368873529136181,
-0.032015688717365265,
0.018252898007631302,
0.10814154148101807,
-0.11695381999015808,
-0.002628342481330037,
-0.023946810513734818,
-0.009309018962085247,
-0.10826356709003448,
-0.045623715966939926,
-0.07078123837709427,
-0.0008918421808630228,
0.005338754970580339,
-0.010726012289524078,
-0.09829964488744736,
-0.009227519854903221,
0.06742778420448303,
-0.05727698653936386,
-0.018329283222556114,
0.05517352744936943,
0.06364213675260544,
-0.05329179763793945,
0.04189195856451988,
-0.026536807417869568,
-0.004791880492120981,
0.0673704445362091,
0.06413023918867111,
-0.04461916536092758,
0.02833482250571251,
-0.024607975035905838,
-0.0546499527990818,
0.009623984806239605,
0.010747595690190792,
-0.07377002388238907,
0.047404300421476364,
-0.08703237771987915,
-0.024844512343406677,
-0.024092257022857666,
0.02308119460940361,
0.09882404655218124,
-0.07226265966892242,
-0.1446578949689865,
0.0006861841538920999,
0.039510902017354965,
0.009328266605734825,
-0.00823440495878458,
6.808484861304286e-34,
-0.040805086493492126,
0.060902081429958344,
-0.019387904554605484,
-0.0005382602685131133,
-0.007150853518396616,
-0.07349088042974472,
0.117100290954113,
0.031287092715501785,
0.021827759221196175,
0.036197539418935776,
0.019504090771079063,
0.03179362788796425,
-0.0575733557343483,
-0.01929977536201477,
-0.01954776979982853,
-0.05587765574455261,
-0.0716104730963707,
-0.034173764288425446,
-0.008994709700345993,
0.01985275372862816,
0.054392099380493164,
-0.027129491791129112,
-0.007565016858279705,
0.01913008838891983,
-0.045932888984680176,
0.01921970583498478,
0.09262502193450928,
-0.03425830975174904,
-0.024422531947493553,
0.04924263805150986,
-0.01177696231752634,
-0.011432988569140434,
-0.007718167267739773,
0.012154928408563137,
0.0024672255385667086,
-0.11913148313760757,
0.03331327065825462,
-0.09496523439884186,
0.057492874562740326,
0.004782006144523621,
0.0558563768863678,
0.020029664039611816,
0.09594498574733734,
0.00481566833332181,
-0.07554604113101959,
0.06967717409133911,
0.0219776201993227,
0.09592253714799881,
-0.011844130232930183,
0.03070865012705326,
0.0016767771448940039,
-0.02797485701739788,
0.0494035966694355,
0.06333104521036148,
0.08843070268630981,
-0.007535872515290976,
-0.045775704085826874,
-0.05438090115785599,
0.018038811162114143,
-0.006676895543932915,
0.0012603271752595901,
-0.028206193819642067,
0.028516873717308044,
0.022309526801109314,
-0.031115597113966942,
-0.012425628490746021,
-0.07162131369113922,
-0.05814739689230919,
0.13872130215168,
0.04696120321750641,
0.053275879472494125,
0.0007802097825333476,
0.0076380460523068905,
-0.06513050943613052,
-0.014194073155522346,
-0.0014436065685003996,
-0.036938972771167755,
0.02353888750076294,
0.061265408992767334,
0.03231819346547127,
-0.005479705985635519,
0.09975086152553558,
-0.03993653878569603,
0.021939409896731377,
-0.03369073569774628,
0.03674209117889404,
-0.0552917942404747,
0.02923537977039814,
0.029092879965901375,
0.039380062371492386,
-0.006426856853067875,
0.019827691838145256,
-0.06755463033914566,
-0.10320694744586945,
-0.02806614525616169,
7.064217349112621e-32,
-0.05465158820152283,
-0.08838637918233871,
-0.03495573252439499,
0.08907991647720337,
0.015715090557932854,
0.026777422055602074,
0.059768225997686386,
-0.03473404794931412,
0.0058556413277983665,
-0.023942530155181885,
0.0605657696723938,
0.0370878167450428,
-0.06959464401006699,
-0.003839812008664012,
-0.01486742403358221,
0.054728567600250244,
0.03724159300327301,
-0.039678480476140976,
-0.014888450503349304,
0.01609652489423752,
-0.015020590275526047,
-0.03521218150854111,
-0.05375037342309952,
0.029954619705677032,
0.03208150342106819,
-0.014712642878293991,
-0.06859765946865082,
0.056659117341041565,
0.0339755117893219,
-0.033535491675138474,
-0.01018733810633421,
0.028471997007727623,
0.054583992809057236,
-0.04619831591844559,
0.07499286532402039,
0.0002632474061101675,
-0.08036032319068909,
0.0034628543071448803,
-0.0029007629491388798,
-0.04164682328701019,
0.0024489189963787794,
0.008238150738179684,
0.024290254339575768,
-0.018949847668409348,
0.012940778397023678,
-0.06767278164625168,
0.11558635532855988,
-0.042866889387369156,
0.023620426654815674,
0.026215579360723495,
-0.0031929260585457087,
0.10256265103816986,
0.030126454308629036,
0.04066004604101181,
-0.029141513630747795,
-0.0037432254757732153,
0.0047617945820093155,
-0.049658406525850296,
-0.008676203899085522,
0.00725272111594677,
0.0889400914311409,
0.09041351079940796,
-0.04152734950184822,
0.03543410077691078
] |
55
Equations and inequalities
1 On a coordinate grid, shade the r
egion that satisfies the inequalities:
y > x β 2, y < 4x and y < 5 β x.
2 On a coordinate grid, shade the r
egion that satisfies the inequalities:
x > β1, y + x < 4, 2x + y < 5 and y > β2.
3 On a coordinate grid, shade the r
egion that satisfies the inequalities:
y > (3 β x)(2 + x) and y + x > 3.
4 On a coordinate grid, shade the r
egion that satisfies the inequalities:
y > x2 β 2 and y < 9 β x2.
5 On a coordinate grid, shade the r
egion that satisfies the inequalities:
y > (x β 3)2, y + x > 5 and y < x β 1.
6 The sketch shows the gr
aphs of the straight lines
with equations:
y = x + 1, y = 7 β x and x = 1.
a Work out the coor
dinates of the points of
intersection of the functions.
b Write down the set of
inequalities that
represent the shaded region shown in the sketch.
7 The sketch shows the gr
aphs of the curves with
equations:
y = 2 β 5x β x2, 2x + y = 0 and x + y = 4.
Write down the set of inequalities that represent the
shaded region shown in the sketch.
8 a On a coordinate grid, shade the r
egion that satisfies
the inequalities
y < x + 4, y + 5x + 3 > 0, y > β1 and x < 2.
b Work out the coor
dinates of the vertices of the shaded region.
c Which of the v
ertices lie within the region identified by the
inequalities?
d Work out the ar
ea of the shaded region.Oy
x
β1β2β12 4 613 5712345678
y = 7 β xy = x + 1x = 1
Oy
xβ1
β2β3 1 β4β5β6 β2 2312345678
β1P
A vertex is only included if
both intersecting lines are included.Problem-solvingExercise 3G
|
[
0.013747014105319977,
-0.003816975513473153,
0.06681542098522186,
-0.0416809543967247,
0.0060602519661188126,
0.015252506360411644,
-0.059428147971630096,
-0.024008916690945625,
-0.1666114628314972,
0.05208682641386986,
-0.09709397703409195,
-0.010302322916686535,
0.03112560696899891,
0.07661455124616623,
0.0008384769898839295,
0.009533477947115898,
-0.05885998532176018,
0.006220927927643061,
-0.054171737283468246,
-0.05310048535466194,
-0.043885212391614914,
-0.10415459424257278,
-0.1079925149679184,
-0.008423862978816032,
-0.003653694177046418,
-0.04792967066168785,
-0.023457560688257217,
-0.05557374656200409,
-0.024999769404530525,
0.004851713310927153,
0.09045855700969696,
0.0042979419231414795,
0.043070100247859955,
-0.07614590227603912,
0.017200415953993797,
-0.01255039218813181,
0.07717437297105789,
0.026710085570812225,
0.0021903207525610924,
-0.05719677358865738,
0.001703705289401114,
0.009406986646354198,
-0.019271112978458405,
0.049296047538518906,
0.03399607166647911,
0.007063582073897123,
0.06518219411373138,
-0.08621755987405777,
0.031668487936258316,
-0.004122416488826275,
0.01250364538282156,
-0.016944196075201035,
-0.0008160432334989309,
0.07182182371616364,
-0.02457352913916111,
-0.07984934747219086,
0.032900985330343246,
-0.03864019736647606,
-0.08831620961427689,
0.052978094667196274,
0.03390716388821602,
-0.019596075639128685,
-0.049372173845767975,
0.057963259518146515,
-0.014735670760273933,
0.023996135219931602,
0.04794776439666748,
-0.017065970227122307,
-0.030957890674471855,
0.11571679264307022,
-0.12584619224071503,
-0.019422216340899467,
-0.07165324687957764,
-0.1178307756781578,
-0.008620659820735455,
0.015951840206980705,
-0.047668229788541794,
-0.024568604305386543,
-0.01132948324084282,
-0.04602915421128273,
0.013162791728973389,
0.04356122761964798,
0.04530356451869011,
0.0493285171687603,
-0.026549771428108215,
-0.017958873882889748,
0.05953824520111084,
0.08272355049848557,
-0.007325970102101564,
0.08798707276582718,
0.0024702020455151796,
0.038058631122112274,
-0.018255848437547684,
-0.05950268357992172,
0.015105458907783031,
-0.1081085056066513,
0.061269763857126236,
-0.0784609317779541,
0.0023070138413459063,
0.04248322546482086,
0.009560593403875828,
-0.051032740622758865,
0.05818558484315872,
-0.04367857053875923,
0.006466992199420929,
-0.008339866064488888,
0.06433454155921936,
-0.009703251533210278,
-0.04636567085981369,
-0.04749505594372749,
-0.014331155456602573,
-0.04210891202092171,
0.010418360121548176,
0.061003122478723526,
-0.028866156935691833,
-0.05594827979803085,
0.05913260206580162,
0.02102748490869999,
0.04150043800473213,
-0.03883298486471176,
-0.04265555739402771,
0.030896421521902084,
0.04189743474125862,
-0.03366081044077873,
-0.05183029919862747,
-0.05089375376701355,
-0.04518287256360054,
0.007403124589473009,
0.03078334964811802,
-0.052629198879003525,
-0.048267289996147156,
-0.01583697833120823,
0.04720231145620346,
0.04269633814692497,
-0.0028784184250980616,
-0.004403965547680855,
0.008751616813242435,
0.05350903794169426,
-0.01407646294683218,
0.04891917482018471,
-0.03440813347697258,
0.07746490836143494,
0.06812535971403122,
-0.015955204144120216,
-0.014553605578839779,
-0.0489717572927475,
-0.0827278196811676,
0.042009592056274414,
-0.03683416545391083,
-0.0033060696441680193,
-0.004670619964599609,
0.011739306151866913,
0.016313889995217323,
0.12207651883363724,
0.004700986202806234,
-0.027965916320681572,
-0.065303735435009,
0.06493084877729416,
0.03407569229602814,
0.005350777879357338,
-0.03703664615750313,
0.06739357113838196,
0.005012397188693285,
0.05785096436738968,
-0.03199651464819908,
-0.009345158003270626,
0.03730306029319763,
0.055833399295806885,
-0.07181102782487869,
-0.02699216641485691,
0.007490853313356638,
0.038330040872097015,
-0.02786519005894661,
0.058783791959285736,
-0.0010186668951064348,
-0.010369166731834412,
0.08371030539274216,
0.00445523438975215,
-0.09095587581396103,
0.09900245815515518,
-0.032335832715034485,
-0.17756015062332153,
0.009436478838324547,
0.0028431524988263845,
-0.010492289438843727,
-0.06866859644651413,
0.0069660316221416,
0.033455897122621536,
-0.06352946162223816,
-0.042991843074560165,
0.007893497124314308,
-0.15995950996875763,
0.0729672908782959,
0.09302486479282379,
-0.11551060527563095,
-0.05700568109750748,
-0.006235170643776655,
0.05903318151831627,
-0.07042785733938217,
-0.03584805503487587,
0.01082321722060442,
0.03651532530784607,
0.09650997072458267,
0.018989644944667816,
0.05662338808178902,
-0.02225351706147194,
-0.03367459774017334,
-0.04325287789106369,
0.01167788915336132,
0.055328454822301865,
-0.0410647988319397,
0.008668535389006138,
-0.042013052850961685,
0.02300948090851307,
0.011797002516686916,
-0.03954480215907097,
0.06403731554746628,
-0.07084731012582779,
-0.0893627256155014,
0.016839688643813133,
0.030222859233617783,
-0.03911294788122177,
-0.0000825825409265235,
-5.978472432785996e-33,
-0.0775703638792038,
-0.005430252756923437,
-0.08530399948358536,
-0.038912173360586166,
-0.04487815871834755,
-0.05396256595849991,
0.1105242520570755,
0.019802864640951157,
0.06905896961688995,
0.03014455921947956,
0.08120231330394745,
0.03599857538938522,
-0.006334417033940554,
0.049875445663928986,
-0.027827460318803787,
-0.08966976404190063,
-0.09467606991529465,
-0.02478477917611599,
-0.0037637250497937202,
0.0010492209112271667,
0.001249831635504961,
0.005830985493957996,
-0.01889832876622677,
0.03105946071445942,
-0.053767841309309006,
0.06922676414251328,
0.06852350383996964,
-0.0629160925745964,
-0.016824498772621155,
0.047407254576683044,
-0.0042582000605762005,
-0.02324754372239113,
0.010168276727199554,
0.023839501664042473,
0.003961824346333742,
-0.07955370843410492,
0.009885613806545734,
-0.0535263828933239,
0.07400689274072647,
-0.004545149393379688,
0.03542052581906319,
-0.017380211502313614,
0.10731717199087143,
0.023288754746317863,
0.002386688720434904,
-0.013227244839072227,
0.017335960641503334,
0.09551024436950684,
-0.08010628819465637,
0.01210344024002552,
-0.019162466749548912,
-0.03189106285572052,
-0.05707786604762077,
0.04202306643128395,
0.1039794459939003,
-0.0949314758181572,
0.01865559257566929,
-0.010268463753163815,
0.03983362391591072,
-0.016364028677344322,
-0.004131976515054703,
-0.049088820815086365,
0.07596658915281296,
0.051708661019802094,
-0.06039854884147644,
0.0700642392039299,
-0.01824076473712921,
-0.07825051993131638,
0.05337024852633476,
0.027722377330064774,
0.030741024762392044,
-0.09527716040611267,
-0.0005583862075582147,
-0.09855460375547409,
-0.06856011599302292,
0.028835687786340714,
0.023341365158557892,
0.10152972489595413,
0.04154276102781296,
0.007454649079591036,
-0.06417110562324524,
0.08351367712020874,
0.01888863928616047,
0.07094289362430573,
-0.02658923529088497,
0.04526152089238167,
0.048761043697595596,
0.05857286602258682,
0.06903692334890366,
0.02581820636987686,
0.017602864652872086,
0.008477501571178436,
0.00002281265551573597,
-0.057926200330257416,
0.054123103618621826,
7.121575595960994e-32,
0.0009986584773287177,
0.004490176681429148,
-0.029727213084697723,
0.06879062950611115,
-0.019158542156219482,
0.020531469956040382,
0.00447250297293067,
-0.004468085709959269,
-0.04649718850851059,
-0.04291248321533203,
0.08115552365779877,
0.01043819822371006,
-0.04757760837674141,
-0.04284724220633507,
-0.03239700570702553,
0.01349011529237032,
-0.024599706754088402,
0.05724533647298813,
-0.03347872942686081,
-0.02081335335969925,
-0.03716699779033661,
0.02720762975513935,
-0.03062247298657894,
0.06166888773441315,
0.02830834873020649,
0.01170858833938837,
-0.08102987706661224,
-0.04276124760508537,
0.02992422506213188,
0.018850378692150116,
0.03593287616968155,
-0.03545933961868286,
0.01742076501250267,
-0.012372036464512348,
0.07954241335391998,
0.052144166082143784,
-0.11308087408542633,
0.023894447833299637,
-0.04426361247897148,
-0.0526687391102314,
-0.0003621558134909719,
-0.01750946417450905,
0.02947939932346344,
-0.01949562318623066,
0.0171134565025568,
-0.05856689065694809,
0.027396148070693016,
-0.00375952385365963,
0.015964072197675705,
0.003081410424783826,
-0.026945488527417183,
0.13477230072021484,
-0.02807443216443062,
0.07378751039505005,
-0.00460885651409626,
-0.0062207793816924095,
0.017803676426410675,
0.005391097627580166,
-0.018280452117323875,
-0.04318669065833092,
0.015847478061914444,
0.07260014116764069,
-0.09001694619655609,
-0.037752263247966766
] |
56
Chapter 3
1 2kx
β y = 4
4kx + 3y = β2
are two simultaneous equations, where k is a constant.a
Show that
y = β2. (3 marks)
b Find an expression f
or x in terms of the constant k. (1 mark)
2 Solve the simultaneous equa
tions:
x + 2y = 3
x2 β 4y2 = β33 (7 marks)
3 Given the sim
ultaneous equations
x β 2y = 13xy β y
2 = 8
a Show that 5
y2 + 3y β 8 = 0. (2 marks)
b Hence find the pairs (x,
y) for which the simultaneous equations are satisfied. (5 marks)
4 a By eliminating
y from the equations
x + y = 2x
2 + xy β y2 = β1
show that x2 β 6x + 3 = 0. (2 marks)
b Hence, or otherwise solve the sim
ultaneous equations
x + y = 2
x2 + xy β y2 = β1
giving x and y in the form a Β± b ββ―__
6 , where a and b are integers. (5 marks)
5 a Given tha
t 3x = 9y β 1, show that x = 2y β 2. (1 mark)
b Solve the simultaneous equa
tions:
x = 2y β 2
x2 = y2 + 7 (6 marks)
6 Solve the simultaneous equa
tions:
x + 2y = 3x
2 β 2y + 4y2 = 18 (7 marks)
7 The curve and the line giv
en by the equations
kx2 β xy + (k + 1)x = 1
β k __ 2 x + y = 1
where k is a non-zero constant, intersect at a single point.
a Find the value of
k. (5 marks)
b Give the coor
dinates of the point of intersection of the line and the curve. (3 marks)E
E
E
E
E
E
E/PMixed exercise 3
|
[
-0.09015283733606339,
0.14823301136493683,
-0.02453632839024067,
0.050311993807554245,
-0.01488872617483139,
-0.0227879099547863,
-0.0119157200679183,
-0.03125980123877525,
-0.050314657390117645,
-0.024703843519091606,
-0.005999353714287281,
-0.013044092804193497,
0.03844192996621132,
-0.02671974152326584,
0.030214160680770874,
-0.03286191076040268,
-0.06635435670614243,
0.008187179453670979,
-0.052538685500621796,
0.0266894344240427,
0.005608031060546637,
-0.033824559301137924,
-0.0684567466378212,
-0.023560378700494766,
0.03055344894528389,
0.0019097509793937206,
0.006730010267347097,
-0.02477823570370674,
0.04040360823273659,
-0.033526141196489334,
0.03386140614748001,
0.06327704340219498,
-0.020678667351603508,
0.039100099354982376,
0.08483442664146423,
0.039426203817129135,
0.0667501762509346,
-0.018177160993218422,
0.011761248111724854,
-0.08483854681253433,
-0.06234592944383621,
-0.00014640459266956896,
-0.02970297820866108,
-0.012446192093193531,
-0.008580714464187622,
-0.020985716953873634,
-0.04290783032774925,
0.051225095987319946,
0.03897109627723694,
0.0009034121176227927,
0.06482907384634018,
-0.01085021160542965,
-0.07449562847614288,
0.034316517412662506,
0.01740133948624134,
-0.037238724529743195,
0.0026066615246236324,
0.07346434891223907,
-0.059788573533296585,
0.053560007363557816,
-0.00775539968162775,
0.020026858896017075,
-0.02673252485692501,
0.045240115374326706,
0.02433183789253235,
0.05252879858016968,
-0.00021270755678415298,
-0.025589250028133392,
0.02876088209450245,
0.08821940422058105,
-0.1424107849597931,
0.04589042440056801,
-0.07038981467485428,
0.00868767686188221,
0.07390859723091125,
-0.052753202617168427,
-0.023867081850767136,
-0.06358333677053452,
0.008772146888077259,
-0.014256308786571026,
-0.015423065051436424,
-0.026692204177379608,
0.01852663978934288,
0.0012432393850758672,
0.025563493371009827,
0.00197847792878747,
-0.08640669286251068,
-0.05220148712396622,
0.014671793207526207,
-0.024406185373663902,
0.09722281247377396,
0.019592825323343277,
-0.026599209755659103,
-0.045575983822345734,
0.06957089900970459,
-0.06277891248464584,
0.0028887963853776455,
-0.013149934820830822,
0.07073744386434555,
0.161972314119339,
0.07035554945468903,
-0.043313268572092056,
-0.02380027435719967,
0.044868919998407364,
0.004643418826162815,
-0.03436256945133209,
0.08296910673379898,
-0.049764323979616165,
-0.04504377022385597,
-0.03129178285598755,
-0.08563060313463211,
-0.07579544186592102,
-0.01225709356367588,
0.018576771020889282,
0.05627879872918129,
0.026870351284742355,
0.10290065407752991,
-0.012265205383300781,
0.06645321100950241,
-0.10598622262477875,
-0.03763073310256004,
-0.012361869215965271,
0.07878768444061279,
-0.010120345279574394,
-0.006982183549553156,
-0.09511807560920715,
-0.028131838887929916,
0.006503687705844641,
-0.03841189667582512,
-0.05778738856315613,
0.029466085135936737,
-0.04455415904521942,
0.00882432609796524,
-0.06079927086830139,
0.010935001075267792,
0.00512901833280921,
-0.0455746166408062,
0.11643770337104797,
0.02746228687465191,
-0.007892812602221966,
-0.010286924429237843,
0.04659297317266464,
-0.021469561383128166,
-0.006003949325531721,
-0.03479214012622833,
0.009074127301573753,
0.014205684885382652,
-0.00563263101503253,
-0.017993349581956863,
0.021131690591573715,
0.037930116057395935,
-0.027737630531191826,
0.015043198131024837,
0.09282888472080231,
0.010117501020431519,
-0.058263592422008514,
-0.009432745166122913,
-0.057285282760858536,
0.07233809679746628,
0.019653640687465668,
-0.03456534445285797,
0.05657317116856575,
0.020283998921513557,
0.004751281812787056,
-0.00795768667012453,
0.020698249340057373,
0.04956379905343056,
0.06108599901199341,
-0.01623658277094364,
-0.019361676648259163,
0.008372791111469269,
0.03010878898203373,
-0.02315184473991394,
0.08546213805675507,
0.03949416056275368,
0.008816673420369625,
0.12220664322376251,
-0.04498758539557457,
-0.01161972712725401,
0.0717945545911789,
-0.055220142006874084,
-0.05603383108973503,
-0.027506183832883835,
0.050364911556243896,
-0.042941171675920486,
-0.059946052730083466,
-0.000976945273578167,
0.06443352997303009,
-0.08587741106748581,
0.06967522203922272,
0.04402380436658859,
-0.011524171568453312,
0.0299691129475832,
0.024320483207702637,
-0.14022327959537506,
0.003386000171303749,
0.0024521767627447844,
-0.0396062508225441,
-0.07465490698814392,
0.04506600275635719,
0.04135088622570038,
0.019366556778550148,
0.03198859468102455,
-0.03558996319770813,
0.023311367258429527,
-0.037677232176065445,
-0.045539792627096176,
0.046346332877874374,
-0.006821350194513798,
0.09090403467416763,
0.04234099015593529,
0.023812217637896538,
-0.07097872346639633,
-0.035199642181396484,
-0.031143851578235626,
-0.058042798191308975,
0.015753235667943954,
-0.06548609584569931,
-0.04465954378247261,
0.019938569515943527,
-0.0575484074652195,
-0.023390457034111023,
0.06372401863336563,
-1.0534247726191624e-33,
-0.027844490483403206,
0.012459656223654747,
-0.1512499302625656,
-0.041582733392715454,
0.002976643620058894,
-0.01547850389033556,
0.04680844768881798,
-0.07571893185377121,
0.0957794040441513,
0.04931216314435005,
0.047748055309057236,
-0.006405755877494812,
-0.017396118491888046,
0.03661424294114113,
-0.019716685637831688,
-0.050416249781847,
0.023287804797291756,
-0.003752155462279916,
0.04080432653427124,
-0.02639755979180336,
0.01588287763297558,
-0.027887482196092606,
-0.038143571466207504,
-0.041196804493665695,
-0.06857319921255112,
-0.033703286200761795,
0.06655065715312958,
-0.10779964923858643,
-0.0836958959698677,
0.06734371930360794,
0.02391052059829235,
-0.036772068589925766,
0.02207178808748722,
-0.02142348326742649,
0.0022948875557631254,
-0.08545820415019989,
-0.01522678043693304,
-0.013756743632256985,
0.0035961621906608343,
-0.04710902273654938,
0.1144878938794136,
0.00219703302718699,
0.047590844333171844,
-0.08488854020833969,
0.06319181621074677,
0.072332002222538,
0.030563201755285263,
-0.007423372007906437,
-0.011576701886951923,
0.013523293659090996,
0.04729516804218292,
-0.08086744695901871,
-0.042195647954940796,
0.026711363345384598,
0.1569540947675705,
-0.0793154388666153,
0.011675277724862099,
-0.014495381154119968,
0.044109929352998734,
-0.035221345722675323,
0.028521310538053513,
-0.027618611231446266,
0.05756845325231552,
0.015738926827907562,
0.006927886512130499,
-0.020257869735360146,
-0.03295445069670677,
-0.07206087559461594,
0.04825370013713837,
0.010650607757270336,
-0.03655557334423065,
-0.005668490193784237,
-0.12268617004156113,
-0.0830678716301918,
0.016537079587578773,
-0.0025605608243495226,
-0.06419502198696136,
0.04772241413593292,
0.07214588671922684,
-0.03487139195203781,
-0.08646807074546814,
0.05882204324007034,
0.011144458316266537,
-0.006236921530216932,
-0.0853394865989685,
0.04424845799803734,
0.07934220135211945,
0.07756873220205307,
0.02532549947500229,
-0.03477894887328148,
0.04477400705218315,
-0.01901252195239067,
0.0066976239904761314,
-0.09539823979139328,
0.07235255092382431,
9.495467629536991e-32,
-0.010979600250720978,
-0.032236650586128235,
-0.044739145785570145,
0.007456102408468723,
0.04945359751582146,
0.039781320840120316,
-0.02532445266842842,
0.010690195485949516,
0.028257222846150398,
-0.05221101641654968,
0.13871382176876068,
-0.03332453593611717,
-0.07132259756326675,
-0.011009248904883862,
-0.07247748970985413,
0.03056829608976841,
-0.07675162702798843,
0.04025604575872421,
-0.05979962646961212,
-0.0382777601480484,
-0.04774196073412895,
0.02855476550757885,
-0.06103156879544258,
0.06062231585383415,
0.01510525681078434,
0.07675039768218994,
-0.04943273589015007,
0.03605657070875168,
-0.10291485488414764,
0.012258869595825672,
0.08963726460933685,
-0.046115849167108536,
0.058748759329319,
-0.04524313285946846,
0.022161617875099182,
0.0336652547121048,
0.047403525561094284,
0.043564386665821075,
-0.024644749239087105,
-0.0009486735798418522,
-0.05670972540974617,
0.026269670575857162,
0.008885525166988373,
0.024027569219470024,
0.02309582754969597,
0.04362102597951889,
-0.01154247671365738,
-0.14111122488975525,
-0.025652529671788216,
-0.020985035225749016,
-0.06781622767448425,
0.011784043163061142,
0.02745947428047657,
-0.0106142433360219,
0.0026479042135179043,
-0.12050008773803711,
-0.06903824210166931,
-0.01672278344631195,
0.022443505004048347,
-0.01565781980752945,
-0.007696462795138359,
0.04616042599081993,
-0.0576852522790432,
-0.031814608722925186
] |
57
Equations and inequalities
8 A person throws a ba
ll in a sports hall. The height of the ball, h m, h
x
can be modelled in relation to the horizontal distance from the
point it was thrown from by the quadratic equation:
h = β 3 __ 10 x2 + 5 _ 2 x + 3 _ 2
The hall has a sloping ceiling which can be mode
lled with equation
h = 15 __ 2 β 1 _ 5 x.
Determine w
hether the model predicts that the ball will hit the ceiling. (5 marks)
9 Give y
our answers in set notation.
a Solve the inequality 3
x β 8 > x + 13. (2 marks)
b Solve the inequality
x2 β 5x β 14 > 0. (4 marks)
10 Find the set of va
lues of x for which (x β 1)(x β 4) < 2(x β 4). (6 marks)
11 a Use algebr
a to solve (x β 1)(x + 2) = 18. (2 marks)
b Hence, or otherwise, find the set of
values of x for which (x β 1)(x + 2) > 18.
Give your answer in set notation. (2 marks)
12 Find the set of va
lues of x for which:
a 6x
β 7 < 2x + 3 (2 marks)
b 2x2 β 11x + 5 < 0 (4 marks)
c 5 < 20 ___ x (4 marks)
d both 6x
β 7 < 2x + 3 and 2x2 β 11x + 5 < 0. (2 marks)
13 Find the set of va
lues of x that satisfy 8 __ x2 + 1 < 9 __ x , x β 0 (5 marks)
14 Find the values of
k for which kx2 + 8x + 5 = 0 has real roots. (3 marks)
15 The equation 2x2 + 4kx β 5k = 0, where k is a constant, has no real roots.
Prove that k satisfies the inequality β 5 _ 2 < k < 0. (3 marks)
16 a Sketch the gra
phs of y = f(x) = x2 + 2x β 15 and g(x) = 6 β 2x on the same axes. (4 marks)
b Find the coordinates of
any points of intersection. (3 marks)
c Write down the set of
values of x for which f(x) > g(x). (1 mark)
17 Find the set of va
lues of x for which the curve with equation y = 2x2 + 3x β 15 is
below the line with equation y = 8 + 2x. (5 marks)
18 On a coordinate grid, shade the r
egion that satisfies the inequalities:
y > x2 + 4x β 12 and y < 4 β x2. (5 marks)
19 a On a coordinate grid, shade the r
egion that satisfies the inequalities
y + x < 6, y < 2x + 9, y > 3 and x > 0. (6 marks)
b Work out the ar
ea of the shaded region. (2 marks)E/P
E
E
E
E
E
E/P
E
E
E
E/P
|
[
0.12120036780834198,
0.07510519027709961,
0.04953298717737198,
-0.1295887976884842,
-0.017856335267424583,
0.00521425623446703,
-0.00046839952119626105,
-0.0020667442586272955,
-0.08508830517530441,
0.02139967679977417,
-0.008192496374249458,
-0.05544678866863251,
0.11054953932762146,
0.04850732535123825,
-0.048186011612415314,
0.06143324449658394,
0.01815546303987503,
0.0036824068520218134,
-0.06935679912567139,
-0.05234166607260704,
0.019023649394512177,
-0.015322207473218441,
-0.03424225375056267,
-0.007017274387180805,
-0.004465526435524225,
-0.11717730015516281,
-0.018551409244537354,
0.03233357146382332,
-0.07158228754997253,
-0.03954608738422394,
-0.026528963819146156,
-0.027825359255075455,
0.06626448035240173,
-0.023974843323230743,
0.02920330874621868,
0.05591776967048645,
0.06227278336882591,
0.013135268352925777,
-0.08842762559652328,
0.016455665230751038,
-0.016457440331578255,
-0.005304649006575346,
-0.002701043151319027,
-0.03169737011194229,
0.03662629425525665,
0.08477023243904114,
0.005358451511710882,
-0.018410388380289078,
-0.02064538560807705,
0.00423216400668025,
0.03262448310852051,
0.004459970630705357,
-0.008970426395535469,
0.0751098021864891,
0.021086249500513077,
-0.022034846246242523,
0.032939694821834564,
-0.06914662569761276,
0.0012589331017807126,
0.005778743885457516,
-0.012289099395275116,
-0.0038678532000631094,
-0.024450620636343956,
-0.0047635165974497795,
0.048745326697826385,
0.028144115582108498,
-0.058160535991191864,
-0.055586349219083786,
0.04898431897163391,
0.09385990351438522,
-0.07299559563398361,
-0.010364341549575329,
-0.028491927310824394,
-0.11941538751125336,
0.020292459055781364,
-0.0004343604960013181,
-0.05777106434106827,
-0.10084443539381027,
0.012574841268360615,
-0.008425254374742508,
-0.04033295065164566,
-0.08623804152011871,
-0.05219879746437073,
0.06482573598623276,
-0.06736297160387039,
-0.03592372685670853,
-0.01101386547088623,
0.09894715994596481,
-0.03911232948303223,
-0.012277261354029179,
0.0467817485332489,
-0.06149353086948395,
-0.026681216433644295,
0.047807954251766205,
0.044808827340602875,
-0.08196161687374115,
-0.013445978052914143,
-0.11146552115678787,
0.012965372763574123,
0.14103756844997406,
-0.046274758875370026,
0.003966107498854399,
0.04063640907406807,
-0.03243355453014374,
0.03356706351041794,
-0.06458739191293716,
0.0004247906035743654,
0.0682467445731163,
-0.020820558071136475,
-0.027833756059408188,
-0.00407178932800889,
-0.021201424300670624,
0.03509513661265373,
0.12355265766382217,
0.035526927560567856,
-0.038319893181324005,
0.04124436900019646,
-0.011523743160068989,
-0.09659667313098907,
-0.03606431186199188,
-0.019772334024310112,
0.001562835997901857,
0.03966163471341133,
-0.055512845516204834,
-0.03133757412433624,
0.04440143704414368,
0.016726402565836906,
0.08355232328176498,
-0.06348099559545517,
-0.05505429953336716,
0.015747303143143654,
-0.0637163519859314,
0.019857682287693024,
-0.04205348342657089,
-0.013802947476506233,
-0.029202625155448914,
0.011464487761259079,
0.07482962310314178,
-0.0038639793638139963,
-0.05119980126619339,
-0.04617650806903839,
0.005474094767123461,
0.12189406901597977,
0.011531516909599304,
0.019256511703133583,
0.05890296399593353,
-0.08924748003482819,
0.04968103766441345,
-0.013089570216834545,
-0.041081976145505905,
-0.061391089111566544,
0.0037141412030905485,
0.027309181168675423,
0.08156860619783401,
-0.047771770507097244,
-0.07845810800790787,
-0.06187859922647476,
0.027332555502653122,
-0.01098528504371643,
-0.024927735328674316,
-0.12841331958770752,
0.03536317124962807,
0.001032594358548522,
0.06461033225059509,
0.07399840652942657,
0.013101431541144848,
-0.003389138262718916,
-0.06286458671092987,
-0.017360182479023933,
-0.05379890650510788,
0.07696367800235748,
0.03978632017970085,
0.016852522268891335,
0.03570742905139923,
0.0034663802944123745,
0.03247546777129173,
0.04135129973292351,
0.01115956250578165,
-0.014588437043130398,
0.026445502415299416,
0.01952015608549118,
-0.11174941807985306,
0.002896001562476158,
0.039329372346401215,
-0.024376146495342255,
-0.02242571860551834,
-0.038030944764614105,
-0.010217721574008465,
-0.0097282063215971,
0.005106515251100063,
0.021705836057662964,
-0.04133730009198189,
-0.002127554267644882,
0.10721936821937561,
-0.05759566277265549,
-0.04662923887372017,
0.04303957521915436,
0.01580687426030636,
-0.06477930396795273,
-0.010418938472867012,
-0.007930213585495949,
0.032158080488443375,
0.04701000824570656,
0.06948209553956985,
0.03978811949491501,
0.01983720064163208,
0.003346770303323865,
-0.009613088332116604,
-0.0007057683542370796,
0.037887029349803925,
-0.08067571371793747,
0.03441232442855835,
-0.10922052711248398,
-0.07424848526716232,
0.06469479948282242,
-0.057105205953121185,
-0.02531418763101101,
-0.02506352588534355,
-0.08721909672021866,
0.029276371002197266,
-0.023859040811657906,
0.013776279054582119,
0.04130887612700462,
6.73681763396404e-33,
-0.08337762951850891,
0.055141981691122055,
-0.05566670373082161,
-0.12467315793037415,
0.02260146290063858,
-0.05137091875076294,
0.03208168223500252,
-0.022237369790673256,
0.14801941812038422,
0.017559099942445755,
0.0008665176574140787,
-0.0006149409455247223,
-0.01690482534468174,
0.023439357057213783,
0.019861551001667976,
-0.1050923690199852,
-0.01336943544447422,
-0.01084853895008564,
-0.013865654356777668,
0.000004765226549352519,
-0.039227575063705444,
0.0036406517028808594,
-0.016981549561023712,
-0.02253769524395466,
-0.05709948018193245,
0.031133264303207397,
0.024462910369038582,
0.0008061937987804413,
-0.03374921530485153,
0.026741886511445045,
0.011140279471874237,
-0.02458558790385723,
0.01423531211912632,
0.09393169730901718,
0.01921388879418373,
-0.06694387644529343,
0.06091684475541115,
-0.04461154341697693,
-0.006121201440691948,
-0.07109585404396057,
0.05737680569291115,
0.04290807619690895,
0.0036148044746369123,
0.05048240348696709,
0.058955904096364975,
0.01823398843407631,
0.05071881413459778,
-0.0037389425560832024,
-0.009890413843095303,
0.017987826839089394,
0.011186590418219566,
0.008978771977126598,
-0.05871983617544174,
0.011393346823751926,
0.09577300399541855,
-0.0490126758813858,
-0.034283481538295746,
-0.023412376642227173,
0.0673915445804596,
-0.023124318569898605,
0.03917114809155464,
-0.07267455756664276,
0.054257217794656754,
0.09996567666530609,
-0.06476513296365738,
0.047137144953012466,
-0.04332033172249794,
-0.03194401040673256,
-0.062182340770959854,
0.03851256147027016,
-0.05753618851304054,
0.056523606181144714,
0.01505403220653534,
-0.005819123238325119,
-0.05293678492307663,
0.00207188306376338,
-0.019898517057299614,
0.027278363704681396,
0.01077139750123024,
-0.01752386800944805,
-0.033513154834508896,
0.07348358631134033,
0.0869683101773262,
-0.017536478117108345,
-0.05078841745853424,
-0.04060503840446472,
0.02742438018321991,
0.003224979154765606,
0.11524640768766403,
0.04102148115634918,
0.07788984477519989,
0.049467865377664566,
0.02936204895377159,
-0.06775786727666855,
-0.04559420049190521,
6.069015619842052e-32,
-0.07204487174749374,
0.03013945370912552,
-0.020619114860892296,
0.03836415708065033,
0.02707226574420929,
0.08114686608314514,
-0.009230760857462883,
-0.05797380954027176,
-0.029986048117280006,
-0.04564977064728737,
0.12303922325372696,
0.04326574131846428,
-0.05249061807990074,
0.007448200136423111,
-0.04661544784903526,
-0.036929164081811905,
-0.022747859358787537,
0.05520472675561905,
0.03183666989207268,
-0.01251799426972866,
-0.0008722394704818726,
0.04940618574619293,
-0.0068689086474478245,
0.0581207349896431,
0.07345064729452133,
0.049693115055561066,
-0.14996668696403503,
-0.037892621010541916,
-0.007193636614829302,
0.008534453809261322,
0.029736146330833435,
0.02287434972822666,
0.023623498156666756,
-0.034491971135139465,
0.05939208343625069,
0.07296352833509445,
-0.04370001703500748,
-0.02004692703485489,
-0.032377153635025024,
0.030529795214533806,
-0.006526364479213953,
-0.08591927587985992,
0.029731929302215576,
-0.05390927195549011,
0.038652025163173676,
-0.07293517887592316,
0.04012390971183777,
-0.06818736344575882,
0.046040479093790054,
-0.04335783049464226,
-0.07863227277994156,
0.10756966471672058,
0.010608049109578133,
0.017895039170980453,
0.008262433111667633,
0.0394599474966526,
-0.02406415343284607,
0.04030568525195122,
0.05111391469836235,
-0.07564666867256165,
0.08686912059783936,
0.07852049916982651,
-0.06971258670091629,
0.047241393476724625
] |
58
Chapter 3
1 Find the possible values of k for the quadratic equation 2 kx2 + 5kx + 5k β 3 = 0
to have real roots.
2 A strai
ght line has equation y = 2 x β k and a parabola has equation
y = 3 x2 + 2kx + 5 where k is a constant. Find the range of values of k for which
the line and the parabola do not intersect.Challenge
1 Linear simultaneous equations can be solved using elimination or substit ution.
2 Simultaneous equations with one linear and one quadratic equation can hav
e up to two pairs
of solutions. You need to make sure the solutions are paired correctly.
3 The solutions of a pair o
f simultaneous equations represent the points of intersection of their
graphs.
4 For a pair o
f simultaneous equations that produce a quadratic equation of the form
ax2 + bx + c = 0:
β’βb2 β 4ac > 0 two r eal solutions
β’βb2 β 4ac = 0 one real solution
β’βb2 β 4ac < 0 no real solutions
5 The solution of an inequality is the set o
f all real numbers x that make the inequality true.
6 To solve a quadr
atic inequality:
β’β Rearrange βsoβthatβtheβright-hand βsideβofβtheβinequality βisβ0
β’β Solv
eβtheβcorr
esponding βquadratic βequationβtoβfindβtheβcriticalβvalues
β’β
Sketchβtheβgraphβofβtheβquadratic βfunction
β’β Useβyourβsket
chβtoβfindβtheβrequir
edβsetβofβvalues
.
7 The values of
x for which the curve y = f(x) is below the curve y = g(x) satisfy the inequality
f(x) < g(x).
The values of x for which the curve y = f(x) is above the curve y = g(x) satisfy the inequality
f(x) > g(x).
8 y <
f(x) represents the points on the coordinate grid below the curve y = f(x).
y > f(x) represents the points on the coordinate grid above the curve y = f(x).
9 If y
> f(x) or y < f(x) then the curve y = f(x) is not included in the region and is represented by
a dotted line.
If y > f(x) or y < f(x) then the curve y = f(x) is included in the region and is represented by a solid line.Summary of key points
|
[
-0.018340013921260834,
0.10894034057855606,
0.027495136484503746,
0.03568187728524208,
0.010295405052602291,
0.004010218195617199,
-0.040273331105709076,
-0.035356562584638596,
-0.07253070175647736,
0.005796846933662891,
-0.006365366745740175,
-0.02673928439617157,
0.025214320048689842,
0.015133421868085861,
0.027456844225525856,
0.03509204462170601,
-0.04605964198708534,
0.016643621027469635,
-0.08383816480636597,
0.030009763315320015,
-0.046458788216114044,
-0.059601929038763046,
-0.0818871408700943,
-0.008579266257584095,
0.026635151356458664,
-0.09251642972230911,
0.03588993847370148,
-0.03795824944972992,
0.05012427270412445,
-0.01645829528570175,
0.015053924173116684,
0.02402215637266636,
0.003174180630594492,
-0.0023133032955229282,
0.05263494327664375,
-0.022871501743793488,
0.07678930461406708,
0.04175746813416481,
0.03161541745066643,
-0.0713798850774765,
-0.0788106843829155,
-0.014148781076073647,
-0.060577843338251114,
-0.06172075867652893,
0.018126068636775017,
0.04684019088745117,
-0.06946054846048355,
-0.037781067192554474,
0.04652971029281616,
-0.04169327765703201,
-0.030513759702444077,
-0.037011340260505676,
-0.05513937398791313,
0.02797870896756649,
0.022966956719756126,
-0.01951693184673786,
-0.06717327982187271,
-0.03711983561515808,
-0.0433783084154129,
0.1110541895031929,
0.038302838802337646,
0.06244337186217308,
0.014399070292711258,
0.05322745814919472,
0.018098680302500725,
0.06291837245225906,
0.06793198734521866,
-0.014476594515144825,
0.010818151757121086,
0.0693981796503067,
-0.0850868821144104,
0.035318851470947266,
-0.03510493412613869,
-0.023481208831071854,
-0.006303930189460516,
-0.006482027471065521,
-0.06450293958187103,
-0.08777142316102982,
-0.01930549181997776,
-0.011449766345322132,
0.030703941360116005,
-0.028353914618492126,
0.04390762746334076,
0.008153459057211876,
-0.042468685656785965,
-0.02335004135966301,
0.058989256620407104,
0.0723944902420044,
0.001978253247216344,
0.00917506031692028,
0.08272229135036469,
-0.0018976718420162797,
0.0024222147185355425,
-0.05898936837911606,
-0.0069767022505402565,
-0.017682287842035294,
0.03214188665151596,
-0.02814478799700737,
0.031156934797763824,
0.0995752215385437,
0.03034917265176773,
-0.10614901781082153,
-0.019578633829951286,
0.05178144946694374,
0.011829206719994545,
0.03983841836452484,
0.03195672109723091,
0.007976057007908821,
0.004633340518921614,
-0.0010710220085456967,
-0.04826907441020012,
-0.042442262172698975,
-0.004814493004232645,
0.029927939176559448,
0.09288466721773148,
0.0762384682893753,
0.07917823642492294,
-0.048891011625528336,
0.029079025611281395,
-0.11719566583633423,
-0.029922138899564743,
-0.05817372724413872,
0.05827098712325096,
-0.046212535351514816,
-0.013506640680134296,
-0.02169712632894516,
-0.00004020387859782204,
0.003803854575380683,
-0.033195462077856064,
-0.041072290390729904,
0.03556634485721588,
-0.057123761624097824,
-0.04102829471230507,
-0.08419431000947952,
0.014559485949575901,
0.05155940353870392,
-0.02344699390232563,
0.08672642707824707,
-0.015691878274083138,
-0.07666011154651642,
0.016376640647649765,
-0.006642072927206755,
0.059262149035930634,
0.07811557501554489,
0.025223147124052048,
0.025390328839421272,
-0.02667166292667389,
0.06225602701306343,
-0.03351302817463875,
-0.02146170847117901,
0.03629988431930542,
0.022816365584731102,
0.05621282011270523,
0.09413004666566849,
0.003182787913829088,
-0.05413266643881798,
0.002183890901505947,
0.0036649678368121386,
-0.015688231214880943,
0.024431202560663223,
-0.01985081285238266,
0.029187537729740143,
-0.10021378099918365,
0.03869691491127014,
0.00221796752884984,
-0.029942229390144348,
-0.02205335721373558,
0.03312307223677635,
-0.03416188061237335,
-0.04239195957779884,
0.048411015421152115,
0.07838035374879837,
0.007320166099816561,
0.07355187833309174,
0.014934835024178028,
0.024166099727153778,
0.07975710183382034,
0.026249445974826813,
-0.0056168800219893456,
0.05654280632734299,
0.004855846520513296,
0.007107913959771395,
-0.00804051198065281,
-0.015772899612784386,
0.032837338745594025,
-0.08036414533853531,
0.06437260657548904,
-0.017158374190330505,
-0.04794809967279434,
-0.0446765162050724,
-0.0012372961500659585,
-0.04103800654411316,
0.007303709164261818,
0.010383323766291142,
-0.13894811272621155,
-0.014951599761843681,
-0.006291941273957491,
0.036279886960983276,
-0.106673464179039,
-0.03840973600745201,
0.01713242195546627,
-0.0032932886388152838,
0.020593564957380295,
0.02244499884545803,
-0.006616794969886541,
0.02214794233441353,
-0.054602328687906265,
0.025614582002162933,
-0.006720638833940029,
-0.007745811250060797,
-0.03600859269499779,
0.09023799002170563,
-0.05935421213507652,
-0.05881790444254875,
-0.022630704566836357,
-0.09877187013626099,
-0.008253399282693863,
-0.03473902493715286,
-0.13515250384807587,
0.06676362454891205,
-0.10100803524255753,
-0.0077554755844175816,
0.06627857685089111,
-7.050237672342871e-34,
0.060921747237443924,
0.017524776980280876,
-0.13710717856884003,
-0.015529405325651169,
0.04564367234706879,
0.05674521252512932,
-0.018634691834449768,
-0.1245361939072609,
0.06350918114185333,
0.01785740815103054,
0.11410801857709885,
-0.022404981777071953,
-0.001250994624570012,
0.12633982300758362,
-0.056844741106033325,
0.03927445784211159,
-0.017000837251544,
0.026616407558321953,
0.055154573172330856,
-0.03196880966424942,
0.06614015996456146,
0.022487275302410126,
-0.025477377697825432,
-0.052417755126953125,
-0.016737034544348717,
0.05459795147180557,
0.07632862776517868,
-0.01694510504603386,
-0.04410333186388016,
0.07413863390684128,
-0.02212366834282875,
-0.026946552097797394,
0.015317045152187347,
0.03126014396548271,
-0.03399757668375969,
-0.039300400763750076,
-0.022301891818642616,
-0.022941794246435165,
0.040051788091659546,
-0.03996441513299942,
0.03183230757713318,
0.026076074689626694,
0.07124414294958115,
-0.017787303775548935,
0.04912164434790611,
0.08781098574399948,
0.06346649676561356,
0.06821271032094955,
0.015809589996933937,
-0.016253918409347534,
0.05107935518026352,
-0.003661723341792822,
-0.019887173548340797,
0.012434720993041992,
0.09120384603738785,
-0.08590705692768097,
0.009670814499258995,
0.0036460403352975845,
0.03897521644830704,
-0.006733357440680265,
0.05698911100625992,
-0.06860467046499252,
0.08147601038217545,
0.03792461380362511,
0.018242018297314644,
0.04109828174114227,
-0.016164960339665413,
0.030595162883400917,
0.03950393572449684,
-0.009031963534653187,
-0.0609700046479702,
0.010327686555683613,
-0.05126147344708443,
-0.01396460272371769,
-0.03521789610385895,
0.05260153114795685,
-0.14495575428009033,
0.06415500491857529,
0.05285050719976425,
-0.11727999895811081,
-0.06012196093797684,
0.03550422191619873,
0.018736563622951508,
-0.08450129628181458,
-0.08434459567070007,
0.06994837522506714,
-0.0033700999338179827,
-0.01435913611203432,
0.06333334743976593,
-0.02738095447421074,
0.07682144641876221,
-0.06679102033376694,
-0.009435949847102165,
-0.07142937928438187,
0.052638985216617584,
7.079193734889272e-32,
-0.07414169609546661,
0.0009205294772982597,
-0.062167033553123474,
0.05475137010216713,
0.04509669914841652,
0.00013121575466357172,
-0.019639937207102776,
-0.02339022420346737,
-0.011266370303928852,
-0.03025086224079132,
0.10700459778308868,
-0.0578964464366436,
-0.14382337033748627,
-0.012320819310843945,
-0.06019149720668793,
-0.0076830037869513035,
-0.0758146271109581,
0.061796046793460846,
-0.03386487439274788,
0.002147071063518524,
-0.039860423654317856,
-0.01608470268547535,
-0.07287032902240753,
0.0431378111243248,
0.046019505709409714,
0.03301677852869034,
-0.09396648406982422,
0.0010730534559115767,
-0.030710339546203613,
0.021681880578398705,
0.05426981672644615,
0.02127080038189888,
-0.006618875078856945,
-0.0071780006401240826,
0.048064224421978,
0.0543907955288887,
-0.06671424955129623,
0.014337234199047089,
-0.06848270446062088,
-0.041536394506692886,
-0.01064286008477211,
-0.06012238934636116,
-0.052651163190603256,
0.030191980302333832,
-0.032677650451660156,
-0.03740057349205017,
-0.010062605142593384,
-0.16327503323554993,
0.07540369033813477,
0.040498748421669006,
-0.0026895960327237844,
0.11112072318792343,
-0.008718537166714668,
0.05927347391843796,
-0.020235272124409676,
-0.07327647507190704,
-0.010493185371160507,
-0.035628922283649445,
0.03462880849838257,
-0.04618997499346733,
0.01223362609744072,
0.00664846645668149,
-0.07648982107639313,
0.007059973664581776
] |
59
Graphs and
transformations
After completing this chapter you should be able to:
β Sketch cubic gr
aphs β pages 60 β 64
β Sketch quartic graphs β pages 64 β 66
β Sketch reciprocal graphs of the form y = a __ x and y = a __ x2 β pages 66 β 67
β Use intersection points of graphs to solve equations β pages 68 β 71
β Translate graphs β pages 71 β 75
β Stretch graphs β pages 75 β 78
β Transform graphs of unfamiliar functions β pages 79 β 81Objectives
1 Factorise these quadratic expressions:
a x2 + 6x + 5 b x2 β 4x + 3
β GCSE Mathematics
2 Sketch the graphs of the following functions:
a y =
(x + 2)(x β 3) b y =
x2 β 6x β 7
β Section 2.4
3 a Copy and complete the table of values for the
function y
= x3 + x β 2.
xβ2 β1.5 β1β0.5 0 0.5 1 1.5 2
yβ12β6.875 β2β1.375
b Use your table o
f values to draw the graph of
y = x3 + x β 2.
β GCSE Mathematics
4 Solve each pair of simultaneous equations:
a y =
2x b y =
x2
x + y = 7 y = 2x + 1
β Sections 3.1, 3.2Prior knowledge check
Many complicated functions can be understood by transforming simpler functions using stretches, reflections and translations. Particle physicists compare observed results with transformations of known functions to determine the nature of subatomic particles.4
|
[
-0.03661707416176796,
0.017481405287981033,
-0.017667531967163086,
-0.02939244545996189,
-0.053239304572343826,
0.04152761772274971,
-0.07440423220396042,
-0.0017043008701875806,
-0.11775834113359451,
0.04791460931301117,
-0.0443246029317379,
-0.05406191200017929,
-0.04597516357898712,
-0.009220963343977928,
-0.0348263680934906,
0.060706350952386856,
-0.014979490078985691,
0.048011235892772675,
-0.04579101502895355,
-0.0550219789147377,
0.02702941745519638,
-0.041982635855674744,
-0.01649576984345913,
-0.09501096606254578,
0.12680745124816895,
-0.06630925089120865,
-0.008035601116716862,
-0.008099389262497425,
-0.011181308887898922,
-0.05668816715478897,
0.01795203424990177,
0.04295291006565094,
0.04461919143795967,
0.0046133678406476974,
0.04463474825024605,
-0.007799339946359396,
0.09056034684181213,
-0.0007929489947855473,
0.03270358592271805,
-0.04546700045466423,
0.013460299000144005,
0.09053073078393936,
-0.009623618796467781,
0.02047664113342762,
0.11659933626651764,
-0.07363160699605942,
-0.08585740625858307,
-0.05257115885615349,
0.04304870590567589,
-0.0802944153547287,
-0.04753660783171654,
-0.003775776829570532,
-0.02938852086663246,
0.03466714546084404,
0.043474871665239334,
-0.039620522409677505,
0.01287646871060133,
-0.01680908538401127,
0.0005407835706137121,
-0.005938840098679066,
-0.009451874531805515,
0.04211987555027008,
-0.024990761652588844,
0.04199909418821335,
-0.03345225006341934,
0.04701276868581772,
0.0704163908958435,
0.022399358451366425,
-0.042635850608348846,
0.1211637631058693,
-0.06980298459529877,
0.057093486189842224,
-0.10914728045463562,
0.004059312399476767,
0.07347451150417328,
0.004287650343030691,
-0.017821917310357094,
0.0000824617818580009,
-0.07655121386051178,
-0.051933664828538895,
-0.05733554810285568,
-0.007078813388943672,
0.10097154974937439,
0.026725994423031807,
0.020346568897366524,
-0.048109643161296844,
0.028789913281798363,
0.04238487780094147,
-0.01243546325713396,
-0.06025933846831322,
-0.012003162875771523,
0.014582259580492973,
0.0024798999074846506,
-0.1517276018857956,
-0.032339293509721756,
-0.09254644811153412,
-0.03964858502149582,
-0.025484690442681313,
0.0873185247182846,
0.05995824187994003,
0.011529575102031231,
0.04478906840085983,
0.06984174251556396,
0.02431594766676426,
-0.05165022984147072,
-0.06183212995529175,
0.001561862649396062,
-0.050476834177970886,
-0.010923938825726509,
-0.05723320692777634,
-0.03515239804983139,
-0.05110449716448784,
-0.0034000612795352936,
-0.004953539464622736,
0.12090405076742172,
-0.10238058120012283,
0.04915405064821243,
-0.021987617015838623,
0.0451546311378479,
0.026355737820267677,
0.015508592128753662,
0.000955757568590343,
0.06127529591321945,
-0.012710672803223133,
-0.0828978419303894,
-0.028944555670022964,
-0.08697736263275146,
0.04595174640417099,
0.0020755776204168797,
0.004515230190008879,
0.06524915993213654,
0.0450415164232254,
0.06358455121517181,
-0.023656390607357025,
-0.018390880897641182,
-0.09122832864522934,
-0.05393867567181587,
0.06800542026758194,
-0.0496988520026207,
0.05987819656729698,
0.020605774596333504,
0.14520584046840668,
0.0018345139687880874,
0.05292340740561485,
-0.0020065868739038706,
-0.0009751942125149071,
-0.0646267905831337,
-0.04993179813027382,
-0.016050467267632484,
-0.034940723329782486,
-0.0003507875371724367,
0.02281581051647663,
0.008601185865700245,
0.10358146578073502,
0.027012405917048454,
-0.04772696644067764,
-0.012919696979224682,
-0.0008949837647378445,
-0.02047780714929104,
-0.004753903951495886,
-0.02345360442996025,
0.04487567022442818,
-0.08174433559179306,
0.024639753624796867,
0.0016528521664440632,
-0.028654927387833595,
0.000035679400752997026,
0.02818685956299305,
-0.020912649109959602,
0.02565084770321846,
0.0054371352307498455,
-0.06207556650042534,
0.041270699352025986,
0.13167718052864075,
0.016109764575958252,
0.02892374061048031,
0.11558929085731506,
-0.07228861004114151,
-0.08682312816381454,
-0.006906301714479923,
0.02551094815135002,
-0.033670615404844284,
-0.0009023817256093025,
0.052166685461997986,
0.013229770585894585,
-0.04019681736826897,
-0.03408414125442505,
0.029121603816747665,
0.028404470533132553,
-0.004849093966186047,
0.0014713139971718192,
-0.05045416206121445,
-0.021345624700188637,
0.034595590084791183,
-0.11165585368871689,
-0.05271127074956894,
0.02821439877152443,
0.012964019551873207,
-0.15880340337753296,
-0.02899470552802086,
0.05559517815709114,
-0.008897156454622746,
0.05192821845412254,
-0.017593059688806534,
-0.010194323025643826,
-0.0503678098320961,
-0.11942286044359207,
-0.054972365498542786,
-0.051170509308576584,
-0.008623374626040459,
-0.015819571912288666,
-0.01099473051726818,
-0.001855434151366353,
0.023495269939303398,
0.04678057134151459,
-0.08047638088464737,
0.03738555684685707,
-0.04393323138356209,
-0.15457074344158173,
-0.01934652589261532,
0.019265681505203247,
-0.0036604246124625206,
0.06715351343154907,
-1.4580443463450906e-33,
0.008434493094682693,
-0.022839752957224846,
-0.09262640029191971,
0.035863980650901794,
0.02448178455233574,
-0.07394372671842575,
0.08868664503097534,
-0.03614816069602966,
0.08448595553636551,
0.0024011791683733463,
0.06692082434892654,
-0.010079351253807545,
0.01596820540726185,
0.03324068337678909,
-0.07965084910392761,
-0.11324426531791687,
0.020082484930753708,
-0.049045201390981674,
-0.05086561664938927,
-0.03731340914964676,
-0.0038578424137085676,
0.020998511463403702,
0.005867138039320707,
-0.05269502475857735,
-0.03114188089966774,
0.08304302394390106,
0.07309778779745102,
-0.07424795627593994,
-0.03961430862545967,
0.04883749037981033,
-0.030188867822289467,
0.02744070626795292,
0.0008745791274122894,
-0.06482360512018204,
-0.033856235444545746,
-0.01378310564905405,
0.013222886249423027,
-0.04140237346291542,
0.02135918289422989,
0.012263037264347076,
0.0679951161146164,
0.019119735807180405,
0.05201996862888336,
-0.027957936748862267,
0.04198510944843292,
0.06895643472671509,
0.03945697098970413,
0.0506497398018837,
-0.06094921752810478,
-0.05007411539554596,
-0.0023806917015463114,
-0.030195731669664383,
0.0905577540397644,
-0.017093274742364883,
0.13651584088802338,
0.01690010353922844,
0.037304796278476715,
-0.04063238576054573,
0.03508676588535309,
0.002625862369313836,
-0.07672945410013199,
-0.11786490678787231,
-0.020007861778140068,
0.027113132178783417,
-0.0557631216943264,
-0.06723299622535706,
-0.035760488361120224,
-0.070036381483078,
0.004446773324161768,
-0.03220609948039055,
-0.0028363342862576246,
0.030425017699599266,
0.03111465461552143,
-0.04141246899962425,
0.019503779709339142,
-0.024581056088209152,
0.014056374318897724,
0.02686213329434395,
0.014120309613645077,
-0.0531238317489624,
-0.041503239423036575,
-0.03771565482020378,
0.0040107835084199905,
0.02443639375269413,
-0.029714366421103477,
-0.006877865642309189,
0.07452699542045593,
0.0961315780878067,
0.08633360266685486,
-0.06501223146915436,
0.038342367857694626,
-0.015533091500401497,
0.01624455861747265,
0.012301770970225334,
0.029569562524557114,
7.212777500663892e-32,
-0.018244661390781403,
0.07031217217445374,
0.02309482917189598,
0.012938002124428749,
0.04600511118769646,
0.03071625344455242,
0.03460327163338661,
0.010247630067169666,
0.04203573614358902,
0.005097596440464258,
0.07174795120954514,
0.09138087928295135,
-0.018126221373677254,
0.02797597274184227,
-0.03150997310876846,
-0.009148966521024704,
0.0008303647045977414,
0.06391417235136032,
0.041292183101177216,
0.009732436388731003,
-0.11447861045598984,
0.03743704408407211,
0.02850978635251522,
0.07962879538536072,
-0.030578428879380226,
0.028071552515029907,
0.02695433795452118,
-0.07544679194688797,
0.003682662034407258,
0.012284339405596256,
0.01761612482368946,
0.06554285436868668,
0.0004277767729945481,
-0.06936179846525192,
0.07219690829515457,
-0.0004931089351885021,
-0.0410456620156765,
0.04314671829342842,
0.011151068843901157,
0.0010451897978782654,
-0.04040063917636871,
0.011123868636786938,
-0.0009785487782210112,
0.02938457950949669,
-0.016428083181381226,
-0.059284038841724396,
0.041550248861312866,
0.019747000187635422,
0.007414717227220535,
-0.024823343381285667,
0.002508779987692833,
-0.02096869982779026,
0.011637292802333832,
-0.07100149244070053,
0.04752717539668083,
-0.04801526293158531,
-0.05751018971204758,
0.007695953827351332,
0.0009635706664994359,
0.047831375151872635,
-0.01446308009326458,
0.1210632175207138,
-0.06834528595209122,
0.015579202212393284
] |
60
Chapter 4
4.1 Cubic graphs
A cubic function has the form f(x) = ax3 + bx2 + cx + d, where a, b, c and d are real numbers and a is
non-zero.
The graph of a cubic function can take several different forms, depending on the exact nature of the
function.
Oy
x Oy
x Oy
x Oy
x
β If p is a root of the function f( x), then the graph of y = f(x) touches or crosses the x-axis at
the point ( p, 0).
You can sketch the graph of a cubic function by finding the roots of the function.For these two functions a is positive. For these two functions a is negative.
Example 1
Sketch the curves with the following equations and show the points where they cross the coordinate axes.
a
y =
(x β 2)(1 β x)(1 + x) b y =
x(x + 1)(x + 2)
a y = (x β 2)(1 β x )(1 + x )
0 = (x
β 2)(1 β x )(1 + x )
So x = 2, x = 1 or x = β 1
So the curve crosses the x -axis at
(2, 0), (1, 0) and ( β1, 0).
When x = 0, y = β 2 Γ 1 Γ 1 = β 2
So the curve crosses the y -axis at (0, β 2).
x β β , y β β β
x β ββ, y β β
2 1 O
β2β1 xy
b y = x(x + 1)( x + 2)
0 = x(x + 1)( x + 2)
So x = 0, x = β 1 or x = β 2Find the value of y when x = 0.
Check what happens to y for large positive and
negative values of x.
The x3 term in the expanded function would be
x Γ (βx) Γ x = βx3 so the curve has a negative x3
coefficient.
Put y = 0 and solve for x. Explore the graph of
y
= (x β p )(x β q )(x β r ) where p , q and r are
constants using GeoGebra.Online
Put y = 0 and solve for x.
|
[
0.0032486168202012777,
0.03257373720407486,
-0.010477319359779358,
-0.018678398802876472,
0.01271580159664154,
0.016270004212856293,
-0.07692519575357437,
-0.012573858723044395,
-0.01479581743478775,
0.09336422383785248,
0.039575833827257156,
-0.012838655151426792,
-0.08487225323915482,
0.020225809887051582,
0.05094953626394272,
0.008969388902187347,
-0.07210247963666916,
0.027885502204298973,
-0.025020349770784378,
0.03639855235815048,
0.039553504437208176,
-0.03564463555812836,
-0.07058343291282654,
-0.061440903693437576,
0.00926514808088541,
-0.0366334468126297,
0.058914490044116974,
0.026594063267111778,
-0.009029602631926537,
-0.012266555801033974,
0.029112232849001884,
0.04330277815461159,
0.059997882694005966,
-0.015394285321235657,
0.052860770374536514,
-0.0604129396378994,
0.022745469585061073,
-0.014613128267228603,
0.028179388493299484,
-0.01805698126554489,
0.029239583760499954,
0.04756578430533409,
0.060152675956487656,
0.023871850222349167,
0.05964190512895584,
-0.048983149230480194,
0.019638823345303535,
-0.052972570061683655,
0.03763210400938988,
-0.00624775281175971,
0.020961448550224304,
0.033494435250759125,
-0.0735095888376236,
0.03798328712582588,
0.04454522579908371,
-0.041047364473342896,
0.04487849399447441,
-0.06980106979608536,
0.015445614233613014,
-0.000014978595572756603,
0.07418657839298248,
0.06275646388530731,
-0.01813606172800064,
0.011389240622520447,
0.03890873119235039,
0.07484188675880432,
-0.01819263957440853,
-0.06494773179292679,
-0.03627907484769821,
0.0533662885427475,
-0.005778308492153883,
-0.001437854254618287,
-0.06366828083992004,
0.0005803798558190465,
-0.004809974227100611,
-0.02163134701550007,
-0.012218596413731575,
-0.0009430635254830122,
-0.09152521193027496,
0.024311505258083344,
-0.09378377348184586,
0.09661009162664413,
0.05663250759243965,
0.04773672670125961,
-0.02260700985789299,
0.0009581397171132267,
0.05387207120656967,
-0.011826787143945694,
-0.04642872139811516,
-0.008088101632893085,
0.0019664920400828123,
0.09183334559202194,
-0.05759282410144806,
-0.09637340903282166,
-0.06266600638628006,
-0.038062430918216705,
-0.0031688823364675045,
-0.00811216700822115,
-0.004808240104466677,
-0.03510658070445061,
-0.08005890250205994,
0.04890132695436478,
-0.02532663382589817,
0.05707613378763199,
-0.05772138759493828,
0.021092789247632027,
0.030362771824002266,
-0.02477092109620571,
0.0005763226072303951,
-0.03173274174332619,
0.017477182671427727,
0.02761065773665905,
-0.01498421747237444,
-0.012369755655527115,
0.06608791649341583,
-0.10858842730522156,
-0.03150716423988342,
-0.062249310314655304,
0.009956584312021732,
0.0151723837479949,
0.0748557522892952,
-0.026901906356215477,
0.018715040758252144,
-0.03949238732457161,
-0.03314487263560295,
-0.011679732240736485,
-0.03557044267654419,
0.04974007606506348,
0.026859262958168983,
-0.02842678874731064,
0.09914512932300568,
-0.006088629364967346,
-0.014682125300168991,
-0.03569215163588524,
-0.014879224821925163,
-0.02713838405907154,
-0.09475535154342651,
0.10469929128885269,
-0.005091600585728884,
0.022239714860916138,
0.08569709211587906,
0.09044177830219269,
-0.002851199358701706,
0.10109066218137741,
0.009925798512995243,
-0.10791771113872528,
-0.04493335634469986,
-0.016412606462836266,
0.02807270921766758,
-0.05047561600804329,
0.031797703355550766,
0.054602425545454025,
-0.006388556677848101,
0.08813851326704025,
-0.005252140574157238,
-0.08026814460754395,
-0.02366616018116474,
-0.08211264759302139,
0.006754002533853054,
0.041887860745191574,
0.03853093087673187,
-0.018500953912734985,
-0.1219797134399414,
0.08651890605688095,
-0.06579677015542984,
0.03752453252673149,
0.06487616896629333,
0.07075167447328568,
-0.002163046970963478,
-0.028334196656942368,
-0.02560686506330967,
-0.04068470001220703,
0.008975062519311905,
0.0686991959810257,
-0.04856758937239647,
-0.03441907465457916,
0.002050996758043766,
0.0044412375427782536,
0.050712428987026215,
-0.05072489753365517,
0.11372198164463043,
0.03370707109570503,
0.038681596517562866,
-0.021589703857898712,
0.0523909330368042,
-0.013270474970340729,
-0.03681259602308273,
-0.002721199067309499,
0.029939942061901093,
-0.08668337762355804,
0.030242808163166046,
0.01840221881866455,
-0.11554165184497833,
0.054212380200624466,
-0.16228243708610535,
-0.003149969270452857,
0.04457418620586395,
0.04490481689572334,
-0.02834372967481613,
0.10769984871149063,
0.012261156924068928,
-0.017051100730895996,
0.06619185209274292,
0.032912664115428925,
-0.13652148842811584,
0.03996749967336655,
-0.03671426698565483,
-0.013751503080129623,
-0.13289305567741394,
-0.020226554945111275,
-0.13453534245491028,
0.0010020635090768337,
0.0071356575936079025,
-0.013059888035058975,
-0.04260218143463135,
0.005570698995143175,
0.02867935597896576,
-0.010133256204426289,
-0.14118634164333344,
-0.0032152007333934307,
-0.014140610583126545,
-0.010526240803301334,
0.018022572621703148,
1.3433127916269736e-33,
0.01019788533449173,
0.007526184897869825,
-0.013937794603407383,
0.023814735934138298,
-0.05477572977542877,
-0.004536097403615713,
0.051757339388132095,
-0.007542097009718418,
0.03458000347018242,
0.010336263105273247,
0.025909235700964928,
0.08233832567930222,
-0.046377550810575485,
0.08109910786151886,
-0.014045950025320053,
0.07051301747560501,
-0.05706285685300827,
-0.029740605503320694,
-0.06569546461105347,
-0.012369929812848568,
0.03691955655813217,
-0.00957451481372118,
-0.09457730501890182,
-0.03639981150627136,
-0.0022910130210220814,
0.024197032675147057,
0.012693308293819427,
-0.0132183488458395,
0.0011823717504739761,
-0.008966401219367981,
-0.050184670835733414,
0.06407493352890015,
0.011897969990968704,
-0.017593106254935265,
-0.002350377384573221,
0.053886573761701584,
-0.0046973880380392075,
-0.056734342128038406,
0.008926606737077236,
0.003583094570785761,
-0.02358158864080906,
0.038129743188619614,
0.14990979433059692,
-0.006489137187600136,
0.07223521918058395,
0.07717572897672653,
-0.0009035299299284816,
-0.010253959335386753,
-0.03891598433256149,
0.03921274095773697,
-0.10675913095474243,
-0.09089101850986481,
0.060380447655916214,
0.06315796077251434,
0.08805229514837265,
0.016412362456321716,
-0.041978295892477036,
-0.03214346989989281,
-0.0458153560757637,
-0.0003755760262720287,
-0.015499352477490902,
-0.09530177712440491,
-0.0071578542701900005,
0.02881569042801857,
-0.048801489174366,
-0.02073788456618786,
-0.10238640010356903,
-0.0601055771112442,
0.01329509261995554,
0.023300381377339363,
0.01824994757771492,
0.051605015993118286,
-0.03086072765290737,
-0.006489401683211327,
-0.10214533656835556,
-0.037004679441452026,
-0.05022840574383736,
0.03724007308483124,
0.013986545614898205,
-0.09279059618711472,
0.06982351094484329,
0.02017815038561821,
-0.0344206802546978,
0.019338618963956833,
-0.014846780337393284,
-0.00005588261774391867,
0.03578253462910652,
0.10622035712003708,
0.07386524230241776,
0.04971998557448387,
0.03759494796395302,
0.02925490401685238,
-0.10805480182170868,
0.029256215319037437,
-0.02068319171667099,
6.171137279316914e-32,
-0.05386029928922653,
0.027723906561732292,
0.003175350371748209,
-0.009919230826199055,
0.02420082315802574,
-0.06499340385198593,
0.0705769881606102,
-0.046327512711286545,
0.04311183840036392,
-0.03603426367044449,
0.07550899684429169,
0.022481610998511314,
-0.053490571677684784,
0.015359921380877495,
-0.001498156925663352,
0.0682828277349472,
0.07192245125770569,
0.02755884826183319,
0.017703013494610786,
0.008654940873384476,
-0.054469604045152664,
-0.027333742007613182,
-0.04212086647748947,
0.0707000344991684,
0.023595457896590233,
0.00044450873974710703,
0.11825833469629288,
-0.02632838673889637,
0.017753932625055313,
-0.0478416383266449,
0.05552350729703903,
0.09895237535238266,
0.005623193457722664,
-0.02092103660106659,
0.06540118902921677,
0.014007392339408398,
0.0032875665929168463,
-0.07698407024145126,
-0.055315542966127396,
-0.041194021701812744,
0.015686949715018272,
-0.07892654836177826,
-0.07512953132390976,
-0.04232561215758324,
-0.09062102437019348,
-0.04251520335674286,
-0.005856323521584272,
-0.029816584661602974,
0.011538888327777386,
0.03169975429773331,
0.044275298714637756,
-0.03652796521782875,
-0.00598698016256094,
-0.03399200364947319,
0.04493677616119385,
0.0125820841640234,
0.025292683392763138,
-0.044974301010370255,
-0.027738092467188835,
0.04064106196165085,
-0.10710924118757248,
0.0974387526512146,
0.08510147780179977,
0.030815739184617996
] |
61Graphs and transformations
So the curve crosses the x -axis at
(0, 0), ( β1, 0) and ( β2, 0).
x β β , y β β
x β ββ, y β β β
β1 1 β2O xyYou know that the curve crosses the x-axis at
(0, 0) so you donβt need to calculate the y-intercept separately.
Check what happens to y for large positive and negative values of x.
Example 2
Sketch the following curves.
a y =
(x β 1)2(x + 1) b y = x3 β 2x2 β 3x c y = (x β 2)3
a y = (x β 1)2(x + 1)
0 = ( x β 1)2(x + 1)
So x = 1 or x = β 1
So the curve crosses the x -axis at ( β1, 0)
and touches the x -axis at (1, 0).
When x = 0, y = ( β1)2 Γ 1 = 1
So the curve crosses the y -axis at (0, 1).
x β β , y β β
x β ββ, y β β β
y
x O β1 11
b y = x3 β 2x3 β 3 x
= x(x2 β 2x β 3)
= x(x β 3)( x + 1)
0 = x(x β 3)( x + 1)
So x = 0, x = 3 or x = β 1
So the curve crosses the x -axis at (0, 0),
(3, 0) and ( β1, 0).
x β β , y β β
x β ββ, y β β β
y
x O β1 3Put y = 0 and solve for x.
(x β 1) is squared so x = 1 is a βdoubleβ repeated
root. This means that the curve just touches the x-axis at (1, 0).
Find the value of y when x = 0.
Check what happens to y for large positive and negative values of x.
This is a cubic curve with a positive coefficient of x
3 and three distinct roots.The x3 term in the expanded function would
be x Γ x Γ x = x3 so the curve has a positive x3
coefficient.
Check what happens to y for large positive and negative values of x.
x β β, y β β
x = 1 is a βdoubleβ repeated root.
x β ββ, y β ββ
First factorise.
|
[
-0.03504672273993492,
-0.06179558113217354,
-0.020597711205482483,
0.017770398408174515,
-0.03536147251725197,
0.0028470130637288094,
-0.002689863322302699,
-0.011815594509243965,
-0.06821368634700775,
0.03606843948364258,
0.0377005971968174,
-0.05392921715974808,
-0.001179443788714707,
0.00905555672943592,
-0.10937006771564484,
-0.007881379686295986,
0.03474630042910576,
-0.025550654157996178,
-0.015874866396188736,
-0.016636302694678307,
0.05902029201388359,
-0.06154470518231392,
-0.045934055000543594,
-0.04246982932090759,
0.026588160544633865,
-0.05888591706752777,
0.04694809392094612,
-0.013206391595304012,
-0.037033308297395706,
0.023515328764915466,
0.04371749237179756,
-0.04296238347887993,
0.028589701279997826,
0.09548094123601913,
0.07364542037248611,
0.0014307507080957294,
0.028816312551498413,
-0.005252438131719828,
0.018615737557411194,
0.01739000529050827,
0.024138933047652245,
0.07053504139184952,
0.010109338909387589,
-0.019001130014657974,
0.06474477797746658,
-0.11529414355754852,
-0.05392651632428169,
-0.07862405478954315,
0.09286686778068542,
0.015385658480226994,
0.06748770922422409,
0.06428235024213791,
-0.00801158044487238,
0.04905300587415695,
-0.049037039279937744,
0.008715198375284672,
-0.025544770061969757,
-0.03487734869122505,
-0.018713613972067833,
-0.029377704486250877,
-0.0473051555454731,
0.04269394278526306,
-0.04783029481768608,
0.03841933235526085,
-0.03214816376566887,
0.0689643919467926,
0.06680966168642044,
-0.03191081061959267,
-0.021671876311302185,
0.05625733733177185,
-0.06102506071329117,
-0.014136971905827522,
-0.026611199602484703,
-0.05560368299484253,
0.000452402833616361,
-0.04519197717308998,
-0.032342854887247086,
0.05838261544704437,
-0.048781923949718475,
-0.08541727811098099,
-0.09026449918746948,
-0.026267878711223602,
0.09750952571630478,
0.08313392102718353,
-0.010203185491263866,
0.038429081439971924,
0.047293346375226974,
0.03327765688300133,
-0.04865170270204544,
0.052595946937799454,
0.04853644222021103,
0.08646830171346664,
-0.05542070046067238,
-0.07559739053249359,
0.009249590337276459,
-0.06684571504592896,
0.0594274140894413,
-0.048921212553977966,
0.0642148032784462,
0.0637269914150238,
0.028576893731951714,
-0.05924142524600029,
0.0017285775393247604,
0.08461663126945496,
0.02904389053583145,
-0.007550740614533424,
-0.050848379731178284,
0.0009834517259150743,
-0.04239572957158089,
-0.005037352442741394,
0.06323599815368652,
-0.07002389430999756,
0.04577626287937164,
-0.0015044116880744696,
0.14302584528923035,
-0.07508329302072525,
-0.008679306134581566,
0.02221439592540264,
0.0104873301461339,
-0.04865814000368118,
0.06229250878095627,
-0.03461720421910286,
0.05803006514906883,
-0.00010003898933064193,
-0.11271950602531433,
0.06173749640583992,
-0.01734130270779133,
-0.036621253937482834,
-0.013154502958059311,
-0.008772140368819237,
0.05651434138417244,
-0.05642245337367058,
-0.021720848977565765,
0.027243325486779213,
0.026520010083913803,
0.021235020831227303,
-0.08444110304117203,
0.11886393278837204,
0.009785096161067486,
-0.025128645822405815,
0.0032697999849915504,
0.059579428285360336,
0.03218637779355049,
0.002856495091691613,
-0.018992504104971886,
0.004596450366079807,
-0.0594300851225853,
0.04614437744021416,
0.034962404519319534,
-0.01370177697390318,
0.027462942525744438,
-0.03757861256599426,
0.021469296887516975,
0.054154831916093826,
0.004045484121888876,
-0.026113661006093025,
-0.043944887816905975,
-0.021808046847581863,
-0.031767603009939194,
-0.051092639565467834,
-0.04754089564085007,
0.03480864688754082,
-0.07469774037599564,
0.020413456484675407,
0.01457137893885374,
0.03852491453289986,
-0.006832500454038382,
0.05851266533136368,
0.04844610020518303,
0.08281826227903366,
0.0748058632016182,
-0.00327692786231637,
-0.008628436364233494,
0.004399911034852266,
0.051381032913923264,
-0.008692965842783451,
0.05392547324299812,
0.03459332883358002,
-0.0330643430352211,
-0.04289097711443901,
-0.024853017181158066,
-0.058967363089323044,
0.015426844358444214,
0.03367059305310249,
-0.006138189230114222,
-0.08159588277339935,
0.0006366216694004834,
-0.06283379346132278,
0.0047777993604540825,
-0.00821483600884676,
-0.007996230386197567,
-0.05074487254023552,
-0.021990912035107613,
0.12289931625127792,
-0.09722757339477539,
-0.028528617694973946,
0.007721007335931063,
0.018228255212306976,
-0.08800890296697617,
0.03565436974167824,
0.026117008179426193,
0.06180282309651375,
0.08257818222045898,
0.05782156065106392,
0.02384296990931034,
0.020111335441470146,
0.005355692468583584,
0.04596247524023056,
0.004916018806397915,
-0.016707977280020714,
0.046023961156606674,
-0.00559413293376565,
-0.006592173129320145,
0.03772205859422684,
0.008958880789577961,
-0.054314736276865005,
-0.030657174065709114,
0.021155277267098427,
-0.11971265822649002,
-0.0017273102421313524,
-0.010964160785079002,
-0.027249015867710114,
0.039261698722839355,
-1.1525115995537886e-33,
0.04549097269773483,
-0.008628849871456623,
-0.06535442918539047,
-0.02141791209578514,
0.013086964376270771,
-0.05952536687254906,
0.053927551954984665,
0.10069777816534042,
0.08188943564891815,
0.019999731332063675,
0.09282726049423218,
0.03916972130537033,
-0.047281358391046524,
0.004150619730353355,
0.006901598535478115,
-0.03437822684645653,
0.020481843501329422,
-0.0894540324807167,
-0.0368468202650547,
-0.08142708986997604,
0.015210411511361599,
-0.008422649465501308,
0.0416395477950573,
0.009115039370954037,
0.007000180426985025,
0.0021602725610136986,
0.063445545732975,
-0.07667525112628937,
-0.05081199109554291,
0.02026495896279812,
-0.06612902879714966,
-0.022290533408522606,
0.10168694704771042,
0.06777062267065048,
-0.09354693442583084,
0.03955991938710213,
-0.08802890032529831,
-0.07513710856437683,
0.05276673659682274,
0.014173327945172787,
0.06266332417726517,
0.06876273453235626,
0.04468845948576927,
0.03801025450229645,
0.007126232143491507,
0.027486825361847878,
-0.021802963688969612,
0.10959111899137497,
-0.03358130156993866,
0.06255601346492767,
-0.055678222328424454,
-0.05265337973833084,
0.08323332667350769,
0.024155274033546448,
0.14857056736946106,
-0.00619635870680213,
0.0029573882929980755,
-0.06721275299787521,
0.043403737246990204,
-0.10659336298704147,
0.027189210057258606,
-0.03878894820809364,
-0.05228675156831741,
0.06550176441669464,
0.03065718151628971,
-0.007588434964418411,
-0.04589512571692467,
-0.10951822251081467,
-0.01788206957280636,
-0.00026151002384722233,
0.06965582072734833,
0.05760541930794716,
-0.07154988497495651,
-0.04857391491532326,
-0.05280953273177147,
-0.10423204302787781,
-0.032756906002759933,
-0.030405845493078232,
0.06347133219242096,
-0.11004743725061417,
-0.056488484144210815,
0.029918501153588295,
-0.04786201938986778,
0.02558327093720436,
-0.03025495447218418,
-0.03861895576119423,
-0.031853772699832916,
0.0908266007900238,
0.040048833936452866,
0.02200825698673725,
0.014695783145725727,
0.01402097474783659,
-0.1212673932313919,
-0.018167613074183464,
0.028941748663783073,
7.570412252946766e-32,
-0.030637357383966446,
0.0007631444022990763,
0.03832462430000305,
0.034907590597867966,
-0.04158640652894974,
0.04050549864768982,
-0.015613183379173279,
-0.12414935976266861,
0.030573014169931412,
-0.0846421867609024,
0.10227207839488983,
0.07954432815313339,
-0.1163552924990654,
0.06957408785820007,
-0.03651917353272438,
0.010547718964517117,
0.03288881108164787,
-0.02393432892858982,
-0.011888307519257069,
-0.003839161479845643,
-0.07137864828109741,
-0.03436398133635521,
-0.036622900515794754,
0.02148042991757393,
-0.046958040446043015,
-0.0493776798248291,
-0.07211025804281235,
-0.026873260736465454,
-0.05508074164390564,
-0.07736831158399582,
-0.003535799216479063,
0.0000227697346417699,
-0.006592385470867157,
-0.052248675376176834,
0.07329530268907547,
-0.029078073799610138,
-0.0920049175620079,
0.06873854249715805,
-0.02787899784743786,
-0.02745893783867359,
-0.05081360787153244,
0.014858581125736237,
-0.007316358853131533,
0.049441784620285034,
-0.0076932357624173164,
0.024630917236208916,
0.05093050003051758,
0.00948014110326767,
0.04396462067961693,
-0.045360829681158066,
-0.021982340142130852,
0.04199918732047081,
0.03406769037246704,
0.03196685016155243,
0.01585688441991806,
-0.07509738951921463,
-0.08761728554964066,
-0.004578079096972942,
0.03719072788953781,
0.010090311989188194,
-0.050323184579610825,
0.030406109988689423,
-0.05972343310713768,
-0.015384008176624775
] |
62
Chapter 4
Example 3
Sketch the curve with equation y = (x β 1)(x2 + x + 2).
y = ( x β 1)( x2 + x + 2)
0 = ( x β 1)( x2 + x + 2)
So x = 1 only and the curve crosses the
x-axis at (1, 0).
When x = 0, y = ( β1)(2) = β 2
So the curve crosses the y -axis at (0, β 2).
x β β , y β β
x β ββ, y β β β
y
x O
β21The quadratic factor x2 + x + 2 gives no solutions
since the discriminant b2 β 4ac = (1)2 β 4(1)(2) = β7.
β Section 2.5
A cubi c graph could intersect the
x-axis at 1, 2 or 3 points. Watch out
Check what happens to y for large positive and
negative values of x.
You havenβt got enough information y
x
to know the exact shape of the graph. It could also be shaped like this:
1 Sketch the following curves and indicate clearly the points of intersection with the axes:
a y =
(x β 3)(x β 2)(x + 1) b y =
(x β 1)(x + 2)(x + 3)
c y =
(x + 1)(x + 2)(x + 3) d y =
(x + 1)(1 β x)(x + 3)
e y =
(x β 2)(x β 3)(4 β x) f y =
x(x β 2)(x + 1)
g y =
x(x + 1)(x β 1) h y =
x(x + 1)(1 β x)
i y =
(x β 2)(2x β 1)(2x + 1) j y =
x(2x β 1)(x + 3)Exercise 4Ac y = (x β 2)3
0 = ( x β 2)3
So x = 2 and the curve crosses the x -axis
at (2, 0) only.
When x = 0, y = ( β2)3 = β8
So the curve crosses the y -axis at (0, β 8).
x β β , y β β
x β ββ, y β β β
y
x O
β82Check what happens to y for large positive and
negative values of x.
x = 2 is a βtripleβ repeated root.
|
[
-0.06215880811214447,
0.018444951623678207,
-0.059025369584560394,
-0.009412138722836971,
-0.018618078902363777,
0.02903864160180092,
-0.02831733599305153,
0.06893282383680344,
-0.10405362397432327,
0.047036632895469666,
0.03328702226281166,
-0.00296430173330009,
0.014659454114735126,
0.04748493805527687,
-0.05681167170405388,
-0.018941741436719894,
-0.07148487120866776,
-0.04453869163990021,
-0.0069715301506221294,
-0.04054570943117142,
0.04826578497886658,
0.015341092832386494,
-0.09702704846858978,
-0.04800792038440704,
0.053551893681287766,
-0.12364659458398819,
0.011273223906755447,
-0.03396207094192505,
0.026800526306033134,
0.003950031474232674,
0.04451408609747887,
-0.03490295261144638,
0.005809785332530737,
0.0321008637547493,
0.07054217159748077,
0.00986650213599205,
0.06511223316192627,
0.023349180817604065,
0.008914239704608917,
0.020891187712550163,
0.01686924509704113,
0.0669720247387886,
-0.054674334824085236,
0.0007284641033038497,
0.10204584151506424,
-0.053921982645988464,
-0.0027372995391488075,
-0.10670531541109085,
0.0023708418011665344,
-0.014661472290754318,
0.03793700784444809,
0.0482218861579895,
-0.013584203086793423,
0.04990290477871895,
-0.061519235372543335,
0.002954190131276846,
-0.06977864354848862,
0.0011748431716114283,
-0.02737264521420002,
0.026388896629214287,
-0.0013836942380294204,
-0.004926703870296478,
-0.03662363067269325,
0.06348992139101028,
-0.025486016646027565,
0.004127266351133585,
0.03014657273888588,
-0.02158777415752411,
-0.009859750047326088,
0.12056289613246918,
-0.13694362342357635,
-0.07955260574817657,
-0.00973817240446806,
-0.05804470553994179,
0.0238118227571249,
-0.08658197522163391,
-0.04378886520862579,
-0.01669269986450672,
-0.0415772907435894,
-0.060929831117391586,
-0.03965160995721817,
-0.02016778290271759,
0.07811437547206879,
0.08817361295223236,
0.002846432849764824,
-0.0006322391564026475,
0.0467543862760067,
-0.03749806806445122,
0.014073215425014496,
-0.00827416218817234,
0.07523266226053238,
0.002030176343396306,
-0.060256924480199814,
-0.013039731420576572,
0.03861209750175476,
-0.06780295073986053,
0.05097538232803345,
-0.060377851128578186,
0.0532209612429142,
0.07959900796413422,
-0.01490423921495676,
-0.04390941187739372,
-0.02296575717628002,
0.049166589975357056,
0.05314100533723831,
-0.0761183351278305,
0.03786840662360191,
-0.027749937027692795,
-0.05916779488325119,
-0.06105562672019005,
0.05211777240037918,
-0.0802014172077179,
0.04900066927075386,
0.011972550302743912,
0.07844694703817368,
-0.10205385833978653,
-0.013937090523540974,
-0.006538220681250095,
0.01778915338218212,
-0.01945592649281025,
0.03528650477528572,
-0.027777573093771935,
0.012321152724325657,
-0.01448393240571022,
-0.10663379728794098,
0.0026803931687027216,
0.034288663417100906,
0.007235844153910875,
-0.017374910414218903,
-0.10767189413309097,
0.013940301723778248,
-0.09341656416654587,
-0.05940207839012146,
-0.006534588523209095,
0.0040030525997281075,
-0.020467707887291908,
-0.11205984652042389,
0.14185672998428345,
0.01452142558991909,
0.004783767741173506,
-0.06594938784837723,
0.031074823811650276,
0.016207708045840263,
-0.03364170715212822,
0.029725192114710808,
-0.05591924861073494,
0.011926934123039246,
0.05303342267870903,
0.018153764307498932,
0.02926952950656414,
-0.02492280676960945,
-0.033867042511701584,
0.05238538607954979,
0.08655410259962082,
0.06768740713596344,
0.02657054178416729,
0.014383461326360703,
-0.02674655243754387,
0.025727344676852226,
-0.03861132636666298,
-0.08513559401035309,
0.06260950863361359,
-0.07458849251270294,
0.02805255353450775,
0.015894899144768715,
-0.03750791400671005,
0.004651848692446947,
0.043541356921195984,
0.055379465222358704,
-0.010202777571976185,
0.0746302604675293,
0.002843001391738653,
0.04389241337776184,
0.09002557396888733,
0.02543727494776249,
0.0028216582722961903,
0.05319700762629509,
0.06396953016519547,
0.008231301791965961,
0.022429998964071274,
-0.0007341198506765068,
-0.0320739783346653,
0.0012254492612555623,
-0.04593830183148384,
-0.036909110844135284,
0.003611411666497588,
-0.06592229008674622,
0.016673646867275238,
0.06961327791213989,
0.06029633432626724,
-0.027591092512011528,
-0.07845387607812881,
-0.0808897614479065,
0.11294235289096832,
-0.04617491364479065,
-0.05922004580497742,
0.03609807416796684,
0.07268741726875305,
-0.04388358071446419,
0.019970718771219254,
-0.08542106300592422,
0.04367997869849205,
0.037296362221241,
0.002864023670554161,
-0.032850805670022964,
-0.02586675062775612,
-0.004828508943319321,
0.013855453580617905,
-0.03649039939045906,
-0.017711946740746498,
0.05016850680112839,
0.01034517027437687,
-0.05800419673323631,
-0.0651303231716156,
-0.008587529882788658,
-0.005082925781607628,
0.033193089067935944,
-0.008360076695680618,
-0.1511443555355072,
0.06277584284543991,
0.01005670614540577,
-0.009635153226554394,
-0.008307650685310364,
-3.9225607311886475e-33,
-0.04515839368104935,
0.052240513265132904,
-0.09648574143648148,
-0.03917720168828964,
-0.01808743178844452,
0.005082491785287857,
0.049781572073698044,
0.1272759586572647,
0.05728418752551079,
0.02968357317149639,
0.052497148513793945,
0.04407787322998047,
-0.0655030906200409,
0.015449638478457928,
-0.028521781787276268,
-0.09583046287298203,
-0.03689754381775856,
-0.06432541459798813,
0.01310157123953104,
-0.06566701084375381,
-0.009220144711434841,
0.006001549772918224,
-0.036342039704322815,
-0.0024713294114917517,
-0.019654173403978348,
-0.004887903109192848,
0.04502533748745918,
-0.060165904462337494,
0.0009493157849647105,
0.03313394635915756,
-0.054686788469552994,
-0.03894262760877609,
0.0031075445003807545,
0.009174097329378128,
-0.008085486479103565,
-0.008143506944179535,
-0.011341524310410023,
-0.07152082771062851,
0.04015164449810982,
0.00315098837018013,
0.03726024553179741,
0.014093919657170773,
0.09403324127197266,
-0.015101349912583828,
0.007216268684715033,
0.04661155119538307,
0.05406883358955383,
0.059346165508031845,
-0.03207610920071602,
0.04173034429550171,
-0.02716374769806862,
-0.05402405560016632,
0.07777509838342667,
0.02132328599691391,
0.13090673089027405,
0.0028162552043795586,
0.014477330259978771,
-0.09114857017993927,
0.013418102636933327,
-0.11468415707349777,
-0.0067827035672962666,
-0.04341382160782814,
-0.03437644988298416,
0.11810072511434555,
0.0315965935587883,
-0.04187715798616409,
-0.10621295124292374,
-0.09695001691579819,
0.02529747411608696,
-0.014921353198587894,
0.02546478435397148,
0.0687238872051239,
-0.026933075860142708,
-0.11874198168516159,
-0.03310489282011986,
-0.02130810171365738,
-0.007615101523697376,
-0.028669118881225586,
0.05334779620170593,
-0.127416729927063,
-0.02006835862994194,
0.072547547519207,
0.034296683967113495,
0.02562960796058178,
-0.04957835376262665,
0.027126912027597427,
0.01664869859814644,
0.13987703621387482,
0.010248465463519096,
0.07387036085128784,
0.03511805087327957,
-0.029708757996559143,
-0.11154148727655411,
-0.0387951098382473,
0.022104285657405853,
8.366934278030053e-32,
-0.12302333861589432,
-0.07289797812700272,
-0.027187269181013107,
0.022494001314044,
0.0178374070674181,
0.04418383166193962,
0.07956815510988235,
-0.07632790505886078,
0.04191641882061958,
-0.046203505247831345,
0.0353027880191803,
0.03300156444311142,
-0.03583722934126854,
0.039594054222106934,
-0.015294312499463558,
0.03809959813952446,
0.0035212531220167875,
-0.030206656083464622,
-0.03691394254565239,
0.0020851728040724993,
-0.033634766936302185,
-0.027330150827765465,
-0.11273904144763947,
-0.00538576440885663,
0.006020662374794483,
-0.03861185535788536,
-0.018220962956547737,
0.008782545104622841,
0.024980826303362846,
-0.016402054578065872,
-0.004151312168687582,
0.019587162882089615,
0.0683944821357727,
0.005054621025919914,
0.040175456553697586,
-0.05616392940282822,
0.0018973405240103602,
0.041516587138175964,
-0.030033767223358154,
-0.06160934269428253,
0.014370978809893131,
-0.0066326940432190895,
0.014254985377192497,
0.008172730915248394,
-0.020569127053022385,
0.04162648320198059,
0.09902486950159073,
0.07498334348201752,
0.01496365386992693,
0.04192096367478371,
-0.025369569659233093,
0.06689698994159698,
0.06380301713943481,
0.011634215712547302,
0.06969647109508514,
-0.052192702889442444,
-0.05146398767828941,
0.004197062458842993,
0.022374795749783516,
0.0011207540519535542,
-0.027704128995537758,
0.04789033532142639,
0.007457389496266842,
0.017197253182530403
] |
63Graphs and transformations
2 Sketch the curves with the f
ollowing equations:
a y =
(x + 1)2(x β 1) b y = (x + 2)(x β 1)2 c y = (2 β x)(x + 1)2
d y = (x β 2)(x + 1)2 e y = x2(x + 2) f y = (x β 1)2x
g y =
(1 β x)2(3 + x) h y = (x β 1)2(3 β x) i y = x2(2 β x)
j y =
x2(x β 2)
3 Factorise the follo
wing equations and then sketch the curves:
a y =
x3 + x2 β 2x b y = x3 + 5x2 + 4x c y = x3 + 2x2 + x
d y =
3x + 2x2 β x3 e y = x3 β x2 f y = x β x3
g y = 12x3 β 3x h y = x3 β x2 β 2x i y = x3 β 9x
j y =
x3 β 9x2
4 Sketch the following curves and indicate the coordinates of the points where the curves cross the
axes:
a y =
(x β 2)3 b y = (2 β x)3 c y = (x β 1)3 d y = (x + 2)3
e y =
β(x + 2)3 f y = (x + 3)3 g y = (x β 3)3 h y = (1 β x)3
i y = β (x β 2)3 j y = β (x β 1 _ 2 ) 3
5 The graph of y = x3 + bx2 + cx + d is shown opposite, where b, c and d y
1 β2β3 O x
β6
are real constants.
a Find the values of
b, c and d. (3 marks)
b Write down the coor
dinates of the point where the curve
crosses the y-axis. (1 mark)
6 The graph of
y = ax3 + bx2 + cx + d is shown opposite, where a, b, c and d y
x β12 3 O2
are real constants.
Find the values of a, b, c and d. (4 marks)
7 Given tha
t f(x) = (x β 10)(x2 β 2x) + 12x
a Express f(x
) in the form x(ax2 + bx + c) where a, b and c are real constants. (3 marks)
b Hence factorise f(x) complete
ly. (2 marks)
c Sketch the gra
ph of y = f(x) showing clearly the points where the graph intersects
the axes. (3 marks)P
Start by writing the equation in the form y = (x β p)(x β q)(x β r).Problem-solving
P
E
|
[
-0.01981847919523716,
-0.026236642152071,
-0.0863470509648323,
-0.0945146381855011,
-0.017643878236413002,
0.0764419212937355,
-0.005962705239653587,
-0.025070903822779655,
-0.08162569999694824,
-0.032204121351242065,
-0.01351320743560791,
-0.03305774927139282,
0.008042678236961365,
0.012390163727104664,
-0.08824853599071503,
-0.018785471096634865,
-0.07948125898838043,
-0.050856295973062515,
-0.04737338796257973,
-0.10102871805429459,
0.014676518738269806,
-0.10784832388162613,
-0.059176795184612274,
-0.10226631909608841,
0.06784357875585556,
-0.03243884816765785,
-0.0419434979557991,
-0.01765606179833412,
-0.024651097133755684,
-0.05691922456026077,
-0.02184785157442093,
0.006798091344535351,
0.07383735477924347,
0.023875761777162552,
0.08633193373680115,
0.019220713526010513,
-0.008620229549705982,
0.019266357645392418,
0.06834851205348969,
-0.008354194462299347,
-0.05096893012523651,
0.03594439476728439,
0.012181592173874378,
0.001562229124829173,
0.06888936460018158,
0.0007042073993943632,
-0.01999278925359249,
0.029534069821238518,
0.03337618708610535,
0.02479337528347969,
0.060268182307481766,
-0.01377182174474001,
-0.07208571583032608,
0.03285503759980202,
0.02368827536702156,
0.0059611620381474495,
-0.01631017029285431,
0.021987412124872208,
-0.02183133363723755,
0.03907529637217522,
-0.017120307311415672,
0.018624749034643173,
-0.02275884710252285,
0.0444837287068367,
-0.06306519359350204,
0.09722477942705154,
0.10565046221017838,
-0.041140999644994736,
-0.020482756197452545,
0.07995107024908066,
-0.0480395145714283,
0.09631428122520447,
-0.07258924841880798,
-0.0578262135386467,
0.02793343923985958,
0.04149769991636276,
0.013417099602520466,
0.07269105315208435,
-0.05315246433019638,
-0.08239896595478058,
-0.09749051928520203,
-0.06245720013976097,
0.1076013445854187,
0.027130350470542908,
-0.00562072591856122,
0.02780681662261486,
-0.013079126365482807,
-0.016345789656043053,
0.006654121447354555,
0.0588267557322979,
-0.0769425481557846,
0.05716162919998169,
-0.053160395473241806,
-0.07168840616941452,
-0.03801233321428299,
-0.08758999407291412,
0.04453306645154953,
0.0042324368841946125,
0.05973350629210472,
0.07441374659538269,
0.002572831930592656,
0.02817949466407299,
0.11632280796766281,
0.05453037843108177,
-0.038901083171367645,
-0.013696207664906979,
-0.06107901781797409,
-0.0224448349326849,
-0.059506818652153015,
-0.004298255778849125,
-0.04167494550347328,
-0.05322567746043205,
0.06464970111846924,
-0.0073969061486423016,
0.03608187288045883,
-0.09790999442338943,
-0.002739088609814644,
0.004856090061366558,
0.014675994403660297,
-0.013430933468043804,
0.015427041798830032,
0.036592401564121246,
0.059416502714157104,
0.028466086834669113,
-0.059311091899871826,
-0.04837307706475258,
-0.09162906557321548,
0.016335753723978996,
-0.012264416553080082,
0.0031106595415621996,
0.04718209058046341,
-0.009782637469470501,
-0.0030225992668420076,
-0.029728984460234642,
-0.026884332299232483,
0.01904374174773693,
-0.04128878191113472,
0.07852466404438019,
-0.032926734536886215,
-0.005375498440116644,
0.00908418744802475,
0.13231772184371948,
0.057111095637083054,
0.018327070400118828,
0.05328909680247307,
-0.03253559768199921,
-0.11132913827896118,
0.05100387707352638,
0.038354914635419846,
0.030350212007761,
-0.031069345772266388,
0.01811516471207142,
-0.018329216167330742,
0.0962858498096466,
0.012046802788972855,
-0.05722270533442497,
-0.045623112469911575,
-0.011820359155535698,
-0.08085222542285919,
-0.0038224186282604933,
-0.02698751725256443,
0.04046139121055603,
-0.01739075779914856,
0.011856134980916977,
0.08816494792699814,
-0.04985526204109192,
0.03616851940751076,
0.027736345306038857,
-0.024577977135777473,
0.015190129168331623,
0.02780759148299694,
-0.03431329131126404,
0.0005636521382257342,
0.09867316484451294,
0.051666658371686935,
-0.002566921291872859,
0.09378796070814133,
-0.05712888389825821,
-0.05328655615448952,
0.015953030437231064,
0.051488492637872696,
-0.030756276100873947,
0.0031480176839977503,
-0.007113623898476362,
-0.048708174377679825,
-0.03818512335419655,
0.023260949179530144,
0.048257458955049515,
-0.00994839332997799,
0.007267998997122049,
-0.05333452299237251,
0.0037036570720374584,
-0.031014101579785347,
0.051651518791913986,
-0.09249047935009003,
-0.013687369413673878,
0.01584300771355629,
0.04893350228667259,
-0.08546923100948334,
-0.03309261053800583,
0.07196345180273056,
0.0667693242430687,
0.111869677901268,
0.06568293273448944,
0.037249498069286346,
-0.06497013568878174,
-0.08631237596273422,
-0.06720061600208282,
-0.06668407469987869,
0.0056089614517986774,
-0.03537948057055473,
-0.00009228872659150511,
-0.0027583211194723845,
0.021546628326177597,
-0.047858431935310364,
-0.07356464862823486,
0.017932377755641937,
-0.05184473842382431,
-0.06705481559038162,
-0.005003770347684622,
0.04111180827021599,
-0.0764402374625206,
0.051079388707876205,
5.9434353542917995e-33,
-0.00020374928135424852,
0.026259558275341988,
-0.09053893387317657,
0.014291036874055862,
-0.019230080768465996,
-0.07167722284793854,
0.033066749572753906,
0.04101381450891495,
0.04712299630045891,
0.017456991598010063,
0.11834724992513657,
0.02022220753133297,
-0.030710536986589432,
-0.06845401972532272,
-0.05832388997077942,
-0.06315863877534866,
0.03574502468109131,
-0.03720816597342491,
-0.036963678896427155,
0.01857300102710724,
0.021322667598724365,
0.025707747787237167,
0.01641620136797428,
0.007938570342957973,
-0.05114663764834404,
0.061519671231508255,
0.05922044813632965,
-0.06595933437347412,
-0.021566331386566162,
0.0541280172765255,
-0.02178146131336689,
-0.03831632062792778,
0.019997701048851013,
0.028720976784825325,
-0.039032235741615295,
0.02958306297659874,
0.029147649183869362,
-0.001542230136692524,
0.0034946552477777004,
0.032071858644485474,
0.02465463988482952,
0.10857842117547989,
0.012123039923608303,
0.02574697509407997,
-0.02118866518139839,
0.0798291265964508,
0.006490419153124094,
0.07516766339540482,
-0.00898707378655672,
0.0037452084943652153,
-0.0331241711974144,
-0.06311266869306564,
-0.03341522067785263,
-0.03653920441865921,
0.03936600685119629,
0.02323395200073719,
-0.006425154395401478,
-0.04783235117793083,
0.057219985872507095,
-0.07216103374958038,
-0.03728676214814186,
-0.05388384684920311,
-0.07849101722240448,
0.07948818057775497,
-0.04920612648129463,
-0.15063433349132538,
-0.09395725280046463,
-0.04806630685925484,
-0.03481542691588402,
0.03193056210875511,
0.053689975291490555,
0.020762991160154343,
-0.004144052974879742,
-0.03595450893044472,
0.03161722049117088,
-0.08874654769897461,
0.049856480211019516,
0.06186610832810402,
-0.023531349375844002,
-0.016881641000509262,
-0.09823465347290039,
0.03204664587974548,
0.02609751932322979,
0.0161344725638628,
0.01851457729935646,
-0.005087863653898239,
0.09419979155063629,
0.05473487079143524,
0.09919039905071259,
0.022361913695931435,
0.044903021305799484,
0.05840672552585602,
0.03825912997126579,
0.006780271418392658,
-0.003611593274399638,
8.805953206197758e-32,
0.02217528037726879,
0.039259232580661774,
0.010566538199782372,
0.013912691734731197,
0.05843181535601616,
0.0037636784836649895,
0.03673500940203667,
-0.004761705175042152,
0.01812935620546341,
-0.09417501837015152,
0.12909768521785736,
0.11347853392362595,
-0.07645297050476074,
0.07861004769802094,
0.039211779832839966,
-0.002223541494458914,
-0.008715025149285793,
-0.005804603453725576,
0.03394297510385513,
-0.0013817125000059605,
-0.09340944141149521,
-0.035944223403930664,
-0.03619231656193733,
0.03026607260107994,
-0.07055100798606873,
-0.06606852263212204,
-0.05942602455615997,
-0.10805144906044006,
-0.03859741985797882,
-0.023442506790161133,
0.013738760724663734,
0.021436434239149094,
0.030907735228538513,
-0.06322896480560303,
0.050463128834962845,
0.0075377230532467365,
-0.11674891412258148,
0.07317257672548294,
-0.03690672293305397,
0.017808379605412483,
-0.011269645765423775,
-0.0033301690127700567,
-0.011306538246572018,
0.07867482304573059,
0.005660689901560545,
0.06015919893980026,
0.04236707091331482,
0.04780665785074234,
0.04765978083014488,
-0.07692337036132812,
-0.04531612619757652,
0.0005397468339651823,
0.03008096292614937,
-0.07376468181610107,
-0.03756416589021683,
-0.012806834653019905,
-0.02744845300912857,
0.03900981321930885,
0.021535737439990044,
-0.009080921299755573,
-0.04153374210000038,
0.013420653529465199,
-0.0012479659635573626,
0.040881067514419556
] |
64
Chapter 4
Example 4
Sketch the following curves:
a y =
(x + 1)(x + 2)(x β 1)(x β 2) b y =
x(x + 2)2(3 β x) c y = (x β 1)2(x β 3)2
a y = (x + 1)( x + 2)( x β 1)( x β 2)
0 = ( x + 1)( x + 2)( x β 1)( x β 2)
So x = β 1, β2, 1 or 2
The curve cuts the x -axis at ( β2, 0), ( β1, 0),
(1, 0) and (2, 0).
When x = 0, y = 1 Γ 2 Γ ( β1) Γ (β2) = 4.
So the curve cuts the y -axis at (0, 4).
x β β , y β β
x β ββ, y β β
Oy
x21/four.ss01
β1β2We know the general shape of the quartic graph
so we can draw a smooth curve through the points.4.2 Quartic graphs
A quartic function has the form f(x) = ax4 + bx3 + cx2 + dx + e, where a, b, c, d and e are real
numbers and a is non-zero.
The graph of a quartic function can take several different forms, depending on the exact nature of the
function.
y
xy
xy
x
You can sketch the graph of a quartic function by finding the roots of the function.This is a
repeated root.
These roots are distinct.
For this function a is negative.For these two functions a is positive.
Check what happens to y for large positive and negative values of x.Substitute x = 0 into the function to find the coordinates of the y-intercept.Set y = 0 and solve to find the roots of the function. Explore the graph of
y
= (x β p )(x β q )(x β r )(x β s ) where p , q, r
and s are constants using GeoGebra.Online
|
[
-0.026368409395217896,
0.025389019399881363,
-0.037942059338092804,
-0.03609432652592659,
-0.0020098411478102207,
0.04330183193087578,
-0.057650335133075714,
0.037844374775886536,
-0.05502891540527344,
0.05546768009662628,
0.049632325768470764,
-0.027806919068098068,
0.004702458158135414,
-0.03158736601471901,
-0.025643862783908844,
-0.006422041449695826,
-0.11703097075223923,
-0.0010847774101421237,
-0.009088069200515747,
-0.04959801211953163,
0.01954246126115322,
-0.05388008803129196,
-0.07687219232320786,
-0.02957957051694393,
0.10100814700126648,
-0.07157889008522034,
0.009029709734022617,
-0.07513554394245148,
-0.03288286179304123,
-0.08513040095567703,
0.005707310978323221,
0.006435713730752468,
0.0781795009970665,
0.025043105706572533,
0.08313094824552536,
0.021572593599557877,
0.03792658448219299,
0.11707528680562973,
0.03356145694851875,
-0.000040013335819821805,
-0.020470324903726578,
-0.021227791905403137,
-0.036368951201438904,
-0.014103325083851814,
0.09976072609424591,
-0.046422481536865234,
-0.0139654241502285,
-0.029685160145163536,
-0.0046867262572050095,
-0.000652538612484932,
0.00044888531556352973,
0.02126449905335903,
-0.06112891063094139,
0.057532671838998795,
-0.04579582437872887,
0.011968325823545456,
-0.10749012976884842,
0.004409961402416229,
-0.0009051006054505706,
0.07681909948587418,
-0.010367260314524174,
0.023266106843948364,
-0.028698453679680824,
0.017797162756323814,
-0.03504129499197006,
-0.011208818294107914,
0.024401545524597168,
-0.0891929417848587,
0.015318374149501324,
0.13194750249385834,
-0.0772436261177063,
-0.017117556184530258,
-0.018506279215216637,
-0.09679612517356873,
-0.05887461453676224,
-0.07030794769525528,
-0.027859006077051163,
0.047605499625205994,
-0.08754871040582657,
-0.0372137688100338,
-0.11115808039903641,
-0.057277776300907135,
0.11982538551092148,
0.07449743151664734,
-0.003338365349918604,
-0.019244875758886337,
0.06323446333408356,
-0.05659288913011551,
0.0059679001569747925,
-0.028056884184479713,
0.02734188921749592,
0.015114719979465008,
-0.1016782894730568,
-0.04494195431470871,
-0.01583646796643734,
-0.12650613486766815,
0.029227571561932564,
-0.040289923548698425,
0.031095772981643677,
0.09353461116552353,
0.000992308370769024,
0.00864440854638815,
0.031132422387599945,
0.07840994000434875,
0.015747729688882828,
-0.019356435164809227,
0.030814694240689278,
-0.047985706478357315,
-0.008321127854287624,
-0.06877300143241882,
0.01502296794205904,
-0.07633277773857117,
0.011958920396864414,
0.04165741428732872,
0.07825952023267746,
-0.06702352315187454,
0.011613091453909874,
-0.013163777999579906,
0.029815439134836197,
-0.02906983345746994,
0.04401835426688194,
-0.03982112184166908,
0.028332754969596863,
-0.01150768157094717,
-0.019915243610739708,
0.00003710197051987052,
-0.006810792721807957,
0.002557852305471897,
-0.018840305507183075,
-0.07125725597143173,
0.03351845592260361,
-0.051897402852773666,
0.008985818363726139,
-0.03986813873052597,
-0.04941762238740921,
-0.058638911694288254,
-0.08879919350147247,
0.11814787983894348,
-0.054730065166950226,
-0.07350710779428482,
-0.004675372503697872,
0.11798880249261856,
0.05820932239294052,
-0.012927337549626827,
0.03873357176780701,
-0.05191531404852867,
-0.07316786795854568,
0.01808319427073002,
0.02107311226427555,
0.060042329132556915,
-0.03746379539370537,
-0.014343943446874619,
-0.00749980378895998,
0.07867378741502762,
-0.022327784448862076,
-0.07794428616762161,
-0.0005118842818774283,
0.005590368527919054,
0.013756463304162025,
-0.011474954895675182,
-0.03611650690436363,
0.042104847729206085,
-0.07077568024396896,
0.06970907747745514,
0.06361420452594757,
-0.04022856429219246,
-0.019716918468475342,
0.08639980107545853,
0.03993110731244087,
-0.07431388646364212,
0.01082085445523262,
0.02546052634716034,
0.007818096317350864,
0.09049376845359802,
0.03914772719144821,
0.06389852613210678,
0.06801801174879074,
0.06095657870173454,
0.01539185456931591,
0.04572296142578125,
0.05853172391653061,
-0.017843924462795258,
-0.015383409336209297,
-0.04487045854330063,
-0.013808385469019413,
-0.0320865623652935,
-0.024779485538601875,
0.028277935460209846,
0.05483205243945122,
0.01813923381268978,
-0.051121581345796585,
-0.032434675842523575,
-0.08195362985134125,
0.11106070131063461,
-0.06471549719572067,
0.026697780936956406,
-0.010570583865046501,
0.05059990659356117,
-0.1024908572435379,
0.05337340757250786,
-0.004306554328650236,
0.05380719527602196,
0.04064226150512695,
-0.0023884272668510675,
-0.09264932572841644,
0.007687213830649853,
-0.0319780632853508,
0.013057182542979717,
-0.04897551238536835,
0.007676251232624054,
0.03164683282375336,
0.02866266295313835,
-0.034934256225824356,
-0.004482539836317301,
-0.035660143941640854,
-0.025634488090872765,
0.08464429527521133,
-0.0633353590965271,
-0.10501942038536072,
0.04753391817212105,
0.006642995402216911,
-0.0008496844093315303,
0.01573672518134117,
9.985922223203244e-33,
-0.06419755518436432,
0.035533297806978226,
-0.06537336111068726,
-0.005127563141286373,
-0.011639907956123352,
-0.060173045843839645,
0.031959909945726395,
0.04936888813972473,
0.023017622530460358,
0.013068165630102158,
0.05528569594025612,
0.048012807965278625,
-0.05410837382078171,
-0.022958282381296158,
0.024897636845707893,
-0.12995833158493042,
-0.05497104674577713,
-0.021685391664505005,
0.01286291889846325,
-0.03677918016910553,
-0.04093560203909874,
0.021987663581967354,
0.015115607529878616,
-0.035912320017814636,
0.01578272506594658,
0.02239997126162052,
0.03874458372592926,
-0.06506931036710739,
-0.002246349584311247,
0.008076772093772888,
-0.0725102350115776,
-0.050514474511146545,
0.008799085393548012,
0.031663890928030014,
-0.0387655533850193,
-0.0074562933295965195,
-0.03500782698392868,
0.0038700599689036608,
0.012913643382489681,
0.012782922945916653,
0.0019034285796806216,
0.028975753113627434,
0.08933056890964508,
0.06670883297920227,
-0.01468491367995739,
0.05622459203004837,
0.051375217735767365,
0.12884405255317688,
-0.009418746456503868,
0.05457332730293274,
-0.05761454999446869,
-0.02492618001997471,
0.08975011855363846,
0.04415731877088547,
0.06348087638616562,
-0.017366845160722733,
-0.004352531395852566,
-0.04691128432750702,
-0.023772448301315308,
-0.09002955257892609,
-0.04624495282769203,
0.0012864969903603196,
-0.10317860543727875,
0.07690434157848358,
0.03474601358175278,
-0.03100987896323204,
-0.03814515843987465,
-0.08855365216732025,
0.006344515364617109,
0.015203378163278103,
0.05824429169297218,
0.16637206077575684,
-0.035345450043678284,
-0.0991155281662941,
-0.03769331052899361,
-0.034027550369501114,
-0.03363202139735222,
0.013427921570837498,
0.01521326880902052,
-0.00515584135428071,
-0.04861703887581825,
0.03434731066226959,
-0.03858990594744682,
-0.002888722810894251,
0.01725379005074501,
0.013122839853167534,
0.01582525111734867,
0.12993910908699036,
0.07535567879676819,
0.015196267515420914,
0.07492075115442276,
-0.02276219241321087,
-0.024298423901200294,
0.019958872348070145,
0.005421421490609646,
7.32450823870955e-32,
-0.04668965935707092,
-0.0766318067908287,
-0.04473751783370972,
0.06248682364821434,
-0.023825349286198616,
0.07256180047988892,
0.03237754479050636,
0.05712800472974777,
0.04217083379626274,
-0.1004050150513649,
0.04348799213767052,
0.04922262951731682,
-0.1301024705171585,
0.13192808628082275,
-0.03298420459032059,
0.05201644077897072,
0.02509867586195469,
-0.0068372394889593124,
-0.011895216070115566,
0.007000403944402933,
-0.0173195768147707,
-0.044577281922101974,
-0.012010700069367886,
0.05926268547773361,
0.010756908915936947,
-0.01831059530377388,
-0.06789384037256241,
-0.0016691242344677448,
0.03283783793449402,
-0.005957141984254122,
0.013598978519439697,
0.01725882850587368,
0.04277092590928078,
-0.024498941376805305,
0.07279295474290848,
0.0072048925794661045,
-0.04513341933488846,
0.02529839053750038,
0.009048822335898876,
-0.015095465816557407,
0.022765373811125755,
0.004423645790666342,
-0.0017010164447128773,
0.03773607686161995,
0.019861357286572456,
-0.02145927958190441,
0.06209108233451843,
0.02495402656495571,
0.03858272358775139,
0.05942336097359657,
-0.029710175469517708,
0.05100800842046738,
0.0034354825038462877,
0.011202785186469555,
0.05256536230444908,
-0.04214279726147652,
-0.025537744164466858,
-0.05206333473324776,
0.03954952210187912,
-0.00985132809728384,
-0.08700434118509293,
0.1130559891462326,
0.014015008695423603,
0.039524924010038376
] |
65Graphs and transformations
b y = x(x + 2)2(3 β x )
0 = x(x + 2)2(3 β x )
So x = 0, β 2 or 3
The curve cuts the x -axis at (0, 0), ( β2, 0)
and (3, 0)
x β β , y β β β
x β ββ, y β β β
Oy
x β2 3
c y = (x β 1)2(x β 3)2
0 = ( x β 1)2(x β 3)2
So x = 1 or 3
The curve touches the x -axis at (1, 0) and
(3, 0).
When x = 0, y = 9.So the curve cuts the y -axis at (0, 9).
x β β , y β β
x β ββ, y β β
Oy
x9
13The coefficient of x4 in the expanded function will
be negative so you know the general shape of the
curve.
These are both βdoubleβ repeated roots, so the curve will just touch the x-axis at these points.
The coefficient of x4 in the expanded function will
be positive.
There are two βdoubleβ repeated roots.
1 Sketch the following curves and indicate clearly the points of intersection with the axes:
a y =
(x + 1)(x + 2)(x + 3)(x + 4) b y =
x(x β 1)(x + 3)(x β 2)
c y =
x(x + 1)2(x + 2) d y = (2x β 1)(x + 2)(x β 1)(x β 2)
e y =
x2(4x + 1)(4x β 1) f y = β(x β 4)2(x β 2)2
g y = (x β 3)2(x + 1)2 h y = (x + 2)3(x β 3)
i y =
β(2x β 1)3(x + 5) j y = (x + 4)4
2 Sketch the following curves and indicate clearly the points of intersection with the axes:a
y =
(x + 2)(x β 1)(x2 β 3x + 2) b y = (x + 3)2(x2 β 5x + 6)
c y =
(x β 4)2(x2 β 11x + 30) d y = (x2 β 4x β 32)(x2 + 5x β 36) In part f the coefficient
of
x4 will be negative.Hint
Factorise the
qu
adratic factor first.HintExercise 4BThere is a βdoubleβ repeated root at x = β2 so the
graph just touches the x-axis at this point.
|
[
-0.010982220061123371,
0.0007282388978637755,
-0.020355822518467903,
-0.06434937566518784,
-0.02992631494998932,
0.08556810021400452,
-0.05478949472308159,
-0.02541484497487545,
-0.10181556642055511,
0.005921386182308197,
-0.018012797459959984,
-0.08479579538106918,
0.004696281161159277,
-0.046110574156045914,
-0.08018940687179565,
0.00621753977611661,
-0.015015272423624992,
-0.027341919019818306,
-0.01009995024651289,
-0.02772989124059677,
0.002522561466321349,
-0.07912030816078186,
-0.02476302906870842,
-0.10869565606117249,
0.07993431389331818,
-0.06365128606557846,
-0.013834754936397076,
-0.03775772824883461,
-0.050247807055711746,
-0.07239530980587006,
-0.033107031136751175,
-0.004352577496320009,
0.0054766028188169,
-0.01708819717168808,
0.07086099684238434,
-0.01419245358556509,
0.006240303628146648,
0.020859697833657265,
-0.002013808349147439,
0.014045070856809616,
-0.05907021835446358,
0.051304809749126434,
0.005178770981729031,
0.013459268026053905,
0.07921183109283447,
-0.04513781517744064,
-0.02512557990849018,
-0.02768375538289547,
0.04949580505490303,
0.010805771686136723,
0.03351937234401703,
0.05126992240548134,
0.03935530409216881,
0.08456725627183914,
0.012018714100122452,
-0.004535581450909376,
-0.032988663762807846,
0.0536406934261322,
0.001529703033156693,
0.027030616998672485,
0.003946932964026928,
0.10423003882169724,
-0.0018665976822376251,
0.039785999804735184,
-0.06494022160768509,
0.09708023816347122,
0.027239622548222542,
-0.01813601329922676,
0.010699841193854809,
0.07189599424600601,
-0.05793466418981552,
0.04964283108711243,
-0.05280749127268791,
-0.05766754224896431,
0.029690397903323174,
-0.059009943157434464,
-0.03822196647524834,
0.0522463396191597,
-0.029742291197180748,
-0.07964400947093964,
-0.10036146640777588,
-0.023027027025818825,
0.07210343331098557,
0.1039896234869957,
0.0022966619580984116,
0.020230231806635857,
0.0370287150144577,
0.07039489597082138,
-0.06897525489330292,
0.022515814751386642,
-0.030765783041715622,
0.06701406836509705,
-0.014640208333730698,
-0.09614823758602142,
0.00879936944693327,
-0.11400485038757324,
0.015375903807580471,
-0.0021092984825372696,
0.060140687972307205,
0.04711436852812767,
0.012528854422271252,
0.050697628408670425,
0.040078260004520416,
0.053241148591041565,
-0.010496428236365318,
-0.04158959537744522,
-0.036313511431217194,
0.026264570653438568,
-0.03481413051486015,
0.02760864794254303,
-0.05948665738105774,
-0.034118566662073135,
0.02775336243212223,
0.02307240664958954,
0.042458426207304,
-0.06782082468271255,
-0.014039193280041218,
-0.016165999695658684,
-0.004712903872132301,
-0.021773872897028923,
-0.0004522419476415962,
0.019592028111219406,
0.04932212829589844,
-0.004397890064865351,
-0.1142418384552002,
-0.02182168886065483,
0.011745375581085682,
0.01286392193287611,
-0.014590412378311157,
-0.018912257626652718,
0.07915007323026657,
-0.0669211745262146,
0.007432658225297928,
-0.029563525691628456,
-0.0484338216483593,
-0.005144092254340649,
-0.04643266275525093,
0.09366462379693985,
-0.025675805285573006,
-0.02056964673101902,
0.018771175295114517,
0.13352979719638824,
0.05917202681303024,
0.04216767102479935,
0.03616245463490486,
-0.030858952552080154,
-0.1256154179573059,
0.03462539613246918,
-0.025097651407122612,
0.0076511618681252,
0.001007078099064529,
-0.012578506022691727,
0.031240250915288925,
0.10066841542720795,
0.029692290350794792,
-0.12224869430065155,
-0.030568929389119148,
-0.0342455618083477,
-0.06582273542881012,
0.005848417524248362,
-0.000823098816908896,
0.035927437245845795,
-0.04558834806084633,
0.03207860514521599,
0.04702601581811905,
0.021597273647785187,
0.08423322439193726,
0.0156230004504323,
-0.008922217413783073,
0.03347179666161537,
0.03129877895116806,
0.0485459566116333,
-0.04928932338953018,
0.11474724858999252,
0.06451782584190369,
-0.02369869314134121,
0.07566117495298386,
0.01684904284775257,
-0.02032928541302681,
0.037597838789224625,
0.04686226323246956,
0.026439204812049866,
0.03762800619006157,
0.02957509458065033,
-0.05773715674877167,
-0.0699392780661583,
0.016779400408267975,
-0.036143675446510315,
0.020910706371068954,
0.026719029992818832,
-0.0037726780865341425,
0.0011113723739981651,
-0.0042391205206513405,
0.08358234912157059,
-0.09541558474302292,
-0.0356350839138031,
0.03048282116651535,
0.021848373115062714,
-0.1048424020409584,
-0.013997283764183521,
0.0604984275996685,
0.06212690845131874,
0.09102959930896759,
0.03354579955339432,
-0.03246552124619484,
-0.005416904576122761,
-0.09862764179706573,
-0.033102430403232574,
-0.022012626752257347,
-0.006747623905539513,
0.02959054335951805,
0.02501758746802807,
-0.013086291961371899,
-0.009334240108728409,
-0.03602788969874382,
-0.04782802611589432,
0.01902497187256813,
-0.04313340410590172,
-0.1127418652176857,
0.0343325138092041,
-0.026607943698763847,
-0.07141795754432678,
0.043261587619781494,
-5.930843605742585e-33,
0.02722741663455963,
0.06250610947608948,
-0.06986460834741592,
-0.04888150468468666,
0.0017501215916126966,
-0.11739495396614075,
0.05908868834376335,
0.012888920493423939,
0.05489078164100647,
0.0161970816552639,
0.09074295312166214,
-0.01998223178088665,
-0.04255779832601547,
-0.005827162880450487,
-0.010800075717270374,
-0.0420655831694603,
-0.00039157478022389114,
-0.07593290507793427,
-0.029347190633416176,
-0.06127096712589264,
0.020611701533198357,
-0.06548810750246048,
-0.015086065046489239,
0.027324240654706955,
-0.00035275286063551903,
0.03720715269446373,
0.030523153021931648,
-0.07162543386220932,
-0.012835840694606304,
0.03154508396983147,
-0.05471562594175339,
-0.0502605140209198,
0.06066844239830971,
0.08179730176925659,
-0.05832141637802124,
0.012058568187057972,
0.008633939549326897,
-0.041595689952373505,
0.09787783026695251,
0.06700565665960312,
0.03767404705286026,
0.04469696804881096,
0.03710322082042694,
0.02925434522330761,
-0.029226239770650864,
0.0667816698551178,
0.04024632275104523,
0.10675860941410065,
-0.03932779282331467,
0.028330091387033463,
-0.04407224804162979,
-0.08644839376211166,
-0.00930716097354889,
0.01845051534473896,
0.1444627195596695,
-0.013751380145549774,
-0.05357649177312851,
-0.01664605736732483,
0.053677789866924286,
-0.11087046563625336,
0.026070486754179,
-0.05182613432407379,
-0.06336947530508041,
0.07905282825231552,
-0.023133397102355957,
-0.09627377241849899,
-0.04835914820432663,
-0.10606491565704346,
-0.0440513901412487,
-0.05037657544016838,
0.06762228161096573,
0.03142501413822174,
-0.032075729221105576,
-0.12164632976055145,
-0.02348352037370205,
-0.07367567718029022,
-0.0003083009214606136,
0.028430938720703125,
0.03221815824508667,
-0.06028738617897034,
-0.07413085550069809,
0.019379347562789917,
0.027755530551075935,
0.06714878231287003,
-0.03641176223754883,
-0.008586439304053783,
0.07688231021165848,
0.0759807825088501,
0.07926953583955765,
0.06346921622753143,
0.021655766293406487,
-0.031691066920757294,
-0.03712575137615204,
-0.0033091565128415823,
-0.002899471204727888,
8.144286244029506e-32,
-0.014923183247447014,
0.013358931057155132,
0.01495062094181776,
0.03462748974561691,
0.024663245305418968,
0.0067829350009560585,
0.017850330099463463,
-0.030789386481046677,
0.013523364439606667,
-0.06086140125989914,
0.05941929295659065,
0.06374450027942657,
-0.12886805832386017,
0.08236204087734222,
-0.01660846173763275,
0.018274907022714615,
0.0161337498575449,
-0.052994538098573685,
0.022112125530838966,
-0.032976023852825165,
-0.11139442026615143,
-0.004449890460819006,
0.005129714030772448,
0.023622214794158936,
-0.08219282329082489,
-0.058186303824186325,
-0.08778806775808334,
-0.05273230001330376,
-0.06105804070830345,
-0.06518042087554932,
-0.014309488236904144,
0.05347567796707153,
0.003343042219057679,
-0.02224459871649742,
0.0632450133562088,
0.01006955374032259,
-0.10877147316932678,
0.06570758670568466,
-0.005589327774941921,
-0.00164405710529536,
0.00854901596903801,
-0.00019145081751048565,
-0.04749395325779915,
0.0567176453769207,
-0.02612309344112873,
-0.012254252098500729,
0.03633664548397064,
0.05912864953279495,
0.017256734892725945,
-0.06339680403470993,
-0.009983779862523079,
0.0329919271171093,
0.06957235187292099,
-0.062133558094501495,
0.023510904982686043,
-0.020264970138669014,
-0.03908857703208923,
0.06129710376262665,
-0.028575051575899124,
0.03456537425518036,
-0.05042629316449165,
0.04407496750354767,
-0.030507782474160194,
0.037052322179079056
] |
66
Chapter 4
3 The graph of
y = x 4 + bx3 + cx2 + dx + e is shown opposite,
where b, c, d and e are real constants.
a Find the coordinates of
point P. (2 marks)
b Find the values of
b, c, d and e. (3 marks)
4 Sketch the gra
ph of y = (x + 5)(x β 4)(x2 + 5x + 14). (3 marks)E/P
OPy
x32 β2β1
E/P
Consider the discriminant of the quadratic factor.Problem-solving
4.3 Reciprocal graphs
You can sketch graphs of reciprocal functions such as y = 1 __ x , y = 1 __ x2 and y = β 2 __ x by considering their
as
ymptotes.
β The graphs of y = k __ x and y = k ___ x2 , where k is a real constant, have asymptotes at x = 0 and
y = 0.
An asy mptote is a line which the
graph approaches but never reaches.Notation
Oy
x Oy
x Oy
x Oy
x1
xy =β2
xy =2
x2y =β5
x2y =
y = k __ x with k > 0. y = k __ x with k < 0. y = k __ x2 with k > 0. y = k __ x2 with k < 0.The graph of y = ax 4 + bx3 + cx2 + dx + e
is shown, where a, b, c, d and e are real
constants.
Find the values of a, b, c, d and e.
Oy
x 3 β13Challenge
|
[
-0.04120328649878502,
0.08240365236997604,
-0.03194872662425041,
-0.04544990882277489,
0.04577682539820671,
0.037999581545591354,
0.01284794881939888,
0.02564185857772827,
-0.03049370087683201,
0.060460373759269714,
0.03172998130321503,
-0.06593326479196548,
-0.04839285835623741,
0.042161427438259125,
-0.06373589485883713,
-0.09244492650032043,
-0.029949095100164413,
0.007522475440055132,
-0.06865322589874268,
-0.014939908869564533,
0.07117322832345963,
-0.06308171898126602,
-0.054621193557977676,
-0.045148324221372604,
0.0808371901512146,
-0.027673998847603798,
0.06915107369422913,
-0.01697923243045807,
0.0041161635890603065,
-0.028192810714244843,
-0.014310103841125965,
-0.013471022248268127,
0.06755482405424118,
0.001111648976802826,
0.06749918311834335,
-0.01078544557094574,
0.05227445438504219,
-0.010182609781622887,
0.06227773055434227,
-0.030849140137434006,
0.03641430661082268,
0.001969030825421214,
-0.08074850589036942,
0.010941830463707447,
0.05053097754716873,
-0.009476770646870136,
-0.07557322829961777,
0.002843621652573347,
0.0745437741279602,
-0.027627956122159958,
0.04113219305872917,
-0.032811544835567474,
-0.08232371509075165,
0.09391667693853378,
-0.015751779079437256,
-0.02519608847796917,
0.05860809609293938,
0.050755809992551804,
0.040255337953567505,
0.07621980458498001,
0.02807736024260521,
0.08358579128980637,
-0.023157119750976562,
0.02033703587949276,
0.02932158298790455,
-0.01917487196624279,
0.06657443195581436,
-0.08602798730134964,
-0.03692392632365227,
0.05720780789852142,
-0.08964933454990387,
-0.015874000266194344,
-0.011116186156868935,
-0.07122105360031128,
0.06843733042478561,
-0.020324410870671272,
-0.07049558311700821,
-0.032996486872434616,
0.0068213073536753654,
-0.058843161910772324,
0.004501092713326216,
0.025018135085701942,
0.07932298630475998,
0.04906497523188591,
0.020838918164372444,
-0.023156020790338516,
0.05599437654018402,
-0.0019566367845982313,
-0.004968215711414814,
0.042250122874975204,
0.02554207295179367,
0.013751232996582985,
-0.06675483286380768,
-0.07728372514247894,
-0.02757970057427883,
-0.08453907072544098,
0.044710516929626465,
-0.05145328491926193,
0.053295381367206573,
0.1276475191116333,
-0.004420906770974398,
0.04178633168339729,
0.007803610526025295,
-0.010938821360468864,
-0.02362036518752575,
-0.008158842101693153,
0.061578523367643356,
-0.04193867743015289,
0.03884311392903328,
-0.09323658794164658,
-0.07150019705295563,
-0.1021742895245552,
0.019490648061037064,
0.05828579142689705,
0.021765058860182762,
-0.05934464558959007,
0.07960225641727448,
-0.039401885122060776,
-0.023674549534916878,
-0.0824010893702507,
0.07052572816610336,
-0.04211391508579254,
-0.0004751902597490698,
-0.030211741104722023,
-0.04838462918996811,
-0.07081697881221771,
-0.05830399692058563,
0.039063211530447006,
-0.005950138904154301,
-0.047412943094968796,
0.12399882078170776,
-0.06906764954328537,
-0.002890749368816614,
-0.040076062083244324,
-0.07482349127531052,
-0.06349847465753555,
-0.01980212889611721,
0.049287375062704086,
-0.030857188627123833,
-0.06115102395415306,
-0.03393488749861717,
0.1387973576784134,
0.001425446942448616,
0.04604966938495636,
-0.020067861303687096,
-0.04219968616962433,
-0.01345368754118681,
-0.021718351170420647,
0.025454530492424965,
0.022936083376407623,
-0.03634711727499962,
0.006890988443046808,
0.006154713220894337,
0.09871426969766617,
0.00019955974130425602,
-0.00884543638676405,
-0.02250358834862709,
-0.02626267448067665,
0.04563815891742706,
-0.02774212881922722,
-0.01894870214164257,
0.02367774397134781,
-0.026401199400424957,
0.11222699284553528,
0.004443211946636438,
-0.02763424441218376,
0.007518913131207228,
0.10394296050071716,
-0.009304163046181202,
0.02131376601755619,
0.023258183151483536,
0.0003181718348059803,
0.017077695578336716,
0.10551215708255768,
0.05410326272249222,
-0.06411100924015045,
0.13322001695632935,
0.067774698138237,
0.011788264848291874,
0.004922030959278345,
-0.03519681468605995,
-0.03844656050205231,
-0.009177697822451591,
0.10019274801015854,
0.01573362573981285,
-0.002673815004527569,
0.014500865712761879,
0.0868184044957161,
-0.026672346517443657,
0.012581138871610165,
0.02145570144057274,
-0.031269531697034836,
-0.05487274006009102,
0.08684025704860687,
-0.15795089304447174,
-0.04865795373916626,
0.026999257504940033,
0.0667479932308197,
-0.0971890538930893,
-0.02234480157494545,
0.07809927314519882,
-0.000992993009276688,
0.034981805831193924,
0.004839781671762466,
-0.024566110223531723,
-0.02063501626253128,
-0.09058492630720139,
0.003811498638242483,
-0.10068117082118988,
0.024158518761396408,
-0.002190622501075268,
-0.0021934586111456156,
-0.03974328190088272,
-0.04775593429803848,
0.022114038467407227,
-0.05118543654680252,
0.07835716754198074,
-0.06473986059427261,
-0.10852600634098053,
-0.02861822582781315,
0.01096117589622736,
-0.05931499972939491,
0.035104334354400635,
1.3738472675800403e-33,
0.0487770140171051,
0.02069946750998497,
-0.061910808086395264,
-0.07246264070272446,
-0.03382333740592003,
0.012392005883157253,
0.05457773059606552,
-0.016183264553546906,
0.08331191539764404,
0.05988103896379471,
0.055922795087099075,
0.0367080494761467,
-0.006834447383880615,
-0.0201320331543684,
-0.02063062973320484,
0.04028279706835747,
-0.08456406742334366,
0.08404561132192612,
-0.013660863973200321,
-0.0632551908493042,
-0.025611376389861107,
0.006957658100873232,
0.031427208334207535,
0.028759479522705078,
0.02007901854813099,
-0.005587685853242874,
0.0771646499633789,
-0.1477564126253128,
-0.04506669566035271,
-0.0373394601047039,
-0.0301959328353405,
-0.025658797472715378,
-0.048670537769794464,
0.001105593517422676,
-0.026638053357601166,
-0.10009602457284927,
-0.016702381893992424,
-0.05089881271123886,
-0.007415632717311382,
-0.001995954429730773,
-0.04217368736863136,
0.07102962583303452,
0.06767897307872772,
-0.02484220266342163,
0.05951607599854469,
0.04195110872387886,
0.113686703145504,
0.008648218587040901,
0.005379976239055395,
0.05942923575639725,
-0.046825796365737915,
-0.004536078777164221,
0.041400179266929626,
0.049251358956098557,
0.08383481949567795,
0.019345413893461227,
0.0436515174806118,
-0.024200839921832085,
-0.03176095336675644,
-0.0698511004447937,
0.003984104376286268,
-0.022603554651141167,
0.020779520273208618,
0.013364214450120926,
-0.050878867506980896,
-0.030367432162165642,
-0.0380813367664814,
0.0025347426999360323,
0.049076713621616364,
-0.003375166794285178,
-0.0028817341662943363,
0.041748445481061935,
-0.04289091005921364,
-0.007096883375197649,
-0.05629436671733856,
-0.03682711720466614,
-0.07226397842168808,
0.03974561020731926,
0.07898838818073273,
-0.004826005548238754,
-0.05692768841981888,
0.0043177856132388115,
0.013018297962844372,
-0.030708342790603638,
-0.03712409362196922,
0.0578666590154171,
0.04286995530128479,
0.05913373455405235,
0.09358717501163483,
-0.04208838939666748,
-0.018137020990252495,
-0.0036690186243504286,
-0.09683912992477417,
0.04016886651515961,
0.050074782222509384,
8.405370004566065e-32,
-0.03979434818029404,
-0.03376515209674835,
-0.02723935805261135,
0.04246097430586815,
0.044063542038202286,
0.002724472898989916,
0.050721339881420135,
-0.0339934378862381,
-0.056168679147958755,
-0.030271343886852264,
0.07701710611581802,
-0.05541504919528961,
-0.07327823340892792,
0.039136163890361786,
-0.0869976207613945,
0.009899375960230827,
-0.04303775727748871,
0.028026055544614792,
-0.0419091060757637,
-0.04948398470878601,
-0.02054019086062908,
-0.0589626207947731,
-0.03251941502094269,
0.022175993770360947,
0.009443596936762333,
0.09526429325342178,
-0.04140263423323631,
-0.04640618711709976,
0.01217698585242033,
-0.03511131927371025,
0.05614297837018967,
-0.02235187217593193,
0.006891068071126938,
-0.004503296222537756,
-0.07089519500732422,
0.0005561117432080209,
0.04234251007437706,
0.027925703674554825,
0.012282204814255238,
0.033099137246608734,
-0.02655196562409401,
-0.039231929928064346,
-0.0382562093436718,
0.006847294978797436,
-0.031096067279577255,
0.04326418787240982,
0.07111169397830963,
-0.08025846630334854,
0.005567594431340694,
0.0041038780473172665,
-0.044172853231430054,
0.057074613869190216,
0.02557472698390484,
0.0005341392825357616,
0.04647228121757507,
-0.08319355547428131,
-0.0595187246799469,
-0.007761856075376272,
-0.07539300620555878,
-0.03103042207658291,
-0.002108685439452529,
0.12680473923683167,
-0.0806596428155899,
0.01966235227882862
] |
67Graphs and transformations
Example 5
Sketch on the same diagram:
a y =
4 __ x and y = 12 ___ x b y = β 1 __ x and y = β 3 __ x c y = 4 __ x2 and y = 10 ___ x2
a
O12
xy =
12
xy =/four.ss01
xy =
/four.ss01
xy =xy
b
c
10
x2 y =10
x2 y =
/four.ss01
x2 y =/four.ss01
x2 y =
x Oy1
xy = β1
xy = β
3
xy = β3
xy = β
x OyThis is a y = k __ x graph with k > 0
In this quadrant, x > 0
so for any values of x: 12 ___ x > 4 __ x
In this quadrant, x
< 0
so for any values of x: 12 ___ x < 4 __ x
This is a y = k __ x graph with k < 0
In this quadrant, x < 0
so for any values of x: β 3 __ x > β 1 __ x
In this quadrant, x
> 0
so for any values of x: β 3 __ x < β 1 __ x
This is a y = k __ x2 graph with k > 0.
x2 is always positive and k > 0 so the y-values are
all positive.
1 Use a separate dia gram to sketch each pair of graphs.
a y =
2 __ x and y = 4 __ x b y = 2 __ x and y = β 2 __ x c y = β 4 __ x and y = β 2 __ x
d y =
3 __ x and y = 8 __ x e y = β 3 __ x and y = β 8 __ x
2 Use a separate dia
gram to sketch each pair of graphs.
a y =
2 __ x2 and y = 5 __ x2 b y = 3 __ x2 and y = β 3 __ x2 c y = β 2 __ x2 and y = β 6 __ x2 Exercise 4C Explore the graph of y = a __ x for
different values of a in GeoGebra.Online
|
[
-0.050240013748407364,
-0.07409496605396271,
-0.04550836980342865,
-0.10134229063987732,
-0.07350761443376541,
0.005849129054695368,
-0.05119254067540169,
-0.04398998245596886,
-0.05289403349161148,
-0.02191075310111046,
-0.032653696835041046,
-0.04588412120938301,
0.03751184418797493,
-0.03143702819943428,
-0.09981934726238251,
0.018743135035037994,
0.00036027110763825476,
0.023605912923812866,
-0.03268113359808922,
-0.005262224934995174,
0.046639334410429,
-0.14639738202095032,
0.025853199884295464,
-0.08247008919715881,
0.09749503433704376,
-0.05674358084797859,
-0.04086985066533089,
-0.022296559065580368,
-0.027308110147714615,
-0.02362081967294216,
-0.06454862654209137,
0.041253622621297836,
0.02855786681175232,
0.032021209597587585,
0.033945564180612564,
-0.05088059976696968,
-0.0043774922378361225,
0.03797135874629021,
0.06933324784040451,
0.041410431265830994,
-0.0566495805978775,
0.0767902061343193,
0.08239124715328217,
-0.016399824991822243,
-0.007254833355545998,
-0.027057290077209473,
-0.030111471191048622,
0.06694666296243668,
0.0681305080652237,
-0.011103047989308834,
0.06417801231145859,
-0.02574794925749302,
-0.022965220734477043,
0.060180529952049255,
-0.03436518833041191,
0.03685253858566284,
0.0312141515314579,
0.07666228711605072,
0.04023969545960426,
-0.08405902236700058,
0.00819904450327158,
0.02348005585372448,
-0.04132544994354248,
-0.004190060775727034,
-0.017617501318454742,
0.046586256474256516,
0.042316682636737823,
-0.04839593917131424,
-0.027221133932471275,
0.022202588617801666,
-0.04579593241214752,
0.09804323315620422,
-0.07516182214021683,
-0.07930412143468857,
0.054141074419021606,
-0.06167158484458923,
0.0024260361678898335,
0.07213979959487915,
-0.043209291994571686,
-0.09914946556091309,
-0.14808650314807892,
-0.0019088893895968795,
0.04827382415533066,
0.025377552956342697,
0.01708846725523472,
0.0007974897162057459,
0.025444963946938515,
-0.029876666143536568,
-0.019055213779211044,
0.042396243661642075,
-0.03159349784255028,
0.046373870223760605,
-0.0012038890272378922,
-0.11650525033473969,
0.04654395580291748,
-0.09849924594163895,
0.014174142852425575,
0.01702956296503544,
0.09359637647867203,
0.04772495850920677,
0.007214280776679516,
0.059466589242219925,
0.11931891739368439,
0.047435078769922256,
-0.0634160190820694,
-0.04994026571512222,
-0.05242809280753136,
-0.0263028834015131,
-0.006286007352173328,
-0.014565921388566494,
-0.07779303938150406,
-0.0374203696846962,
0.009219205938279629,
0.004420868586748838,
-0.03786613047122955,
-0.07932587713003159,
-0.021789656952023506,
0.04112227261066437,
0.03324781358242035,
0.012626837939023972,
0.03717619553208351,
0.02245263382792473,
-0.0012022324372082949,
0.01764300838112831,
-0.05430889502167702,
-0.015045465901494026,
-0.08142060041427612,
0.026620734483003616,
0.008129882626235485,
-0.018843427300453186,
0.07642839848995209,
-0.01808355376124382,
0.05935484170913696,
-0.016258111223578453,
-0.03641830012202263,
0.013735556975007057,
-0.0032647675834596157,
0.086789570748806,
-0.059949006885290146,
-0.006608699448406696,
0.014151106588542461,
0.1172749400138855,
0.05290471762418747,
0.017330849543213844,
-0.0369730144739151,
0.01320898812264204,
-0.07124002277851105,
-0.014123603701591492,
0.012688476592302322,
0.02168520726263523,
0.004641302861273289,
-0.04975522309541702,
-0.02323674038052559,
0.11310799419879913,
-0.05342082306742668,
-0.011957085691392422,
-0.018115907907485962,
-0.036756303161382675,
-0.09353944659233093,
0.019106825813651085,
0.010092721320688725,
0.06518883258104324,
0.020217442885041237,
0.017561785876750946,
0.05898938700556755,
0.03475195914506912,
0.052677396684885025,
0.06158401444554329,
0.0026129630859941244,
0.05714691802859306,
0.026567911729216576,
-0.029334012418985367,
0.0148735037073493,
0.08720126748085022,
0.004214654676616192,
-0.014880922622978687,
0.041029296815395355,
-0.038513731211423874,
-0.053834233433008194,
-0.002735805930569768,
0.023871198296546936,
-0.028760498389601707,
0.006806253921240568,
0.032863542437553406,
-0.058662474155426025,
0.0025133683811873198,
-0.0579187273979187,
-0.004546249285340309,
-0.07801391929388046,
-0.034894466400146484,
0.021625090390443802,
-0.044535811990499496,
0.022616224363446236,
0.031833235174417496,
-0.14185109734535217,
0.0033908840268850327,
0.01101453136652708,
-0.04117186367511749,
-0.09861506521701813,
0.0022891040425747633,
0.07097220420837402,
0.06345300376415253,
0.07351218909025192,
0.03293033316731453,
-0.03343638405203819,
-0.03190818801522255,
-0.02648613043129444,
-0.055728599429130554,
-0.06228930875658989,
0.04016489535570145,
0.00324080977588892,
-0.01985771767795086,
-0.040993582457304,
0.006521907635033131,
0.056874580681324005,
-0.10226358473300934,
0.028155671432614326,
-0.026040339842438698,
0.012903903611004353,
-0.0718914270401001,
-0.02528492547571659,
-0.05041981860995293,
0.0789177194237709,
-2.468073265445252e-33,
0.001866710837930441,
0.02481653168797493,
-0.06043412536382675,
-0.023678049445152283,
0.009005097672343254,
-0.16630873084068298,
0.08089705556631088,
-0.009245557710528374,
0.058038827031850815,
0.005861612968146801,
0.07654987275600433,
-0.028329556807875633,
-0.0880175307393074,
-0.04954208806157112,
-0.04411620274186134,
-0.07711421698331833,
0.05993228778243065,
0.04784046858549118,
-0.028471173718571663,
-0.031251389533281326,
0.039824388921260834,
0.013283612206578255,
0.032603491097688675,
0.041169505566358566,
-0.031786832958459854,
0.05888922140002251,
0.03445814177393913,
-0.058046042919158936,
-0.06530294567346573,
0.03953610360622406,
-0.04251199588179588,
-0.010625224560499191,
0.07042227685451508,
0.05260544270277023,
-0.01886848732829094,
0.0021098197903484106,
-0.029176555573940277,
-0.03307366743683815,
0.01829575002193451,
0.01154016051441431,
0.05945051088929176,
0.09091784805059433,
0.0023507210426032543,
0.04095221683382988,
-0.010257295332849026,
0.06258692592382431,
0.046316977590322495,
0.05839627608656883,
0.004997204057872295,
-0.02225196361541748,
-0.029025482013821602,
-0.044297605752944946,
0.041847772896289825,
-0.06190723553299904,
0.1377997249364853,
0.044457852840423584,
0.03321661427617073,
-0.03352899104356766,
0.06150892749428749,
-0.026913201436400414,
-0.013571013696491718,
-0.03540422394871712,
0.035517916083335876,
-0.0291940625756979,
-0.05859246477484703,
-0.06216185912489891,
-0.05312004312872887,
-0.057736851274967194,
-0.03469082713127136,
0.0009770473698154092,
0.037358783185482025,
0.038759659975767136,
-0.06020571291446686,
-0.05277848243713379,
0.0411844402551651,
-0.08562034368515015,
0.03459412232041359,
-0.012226912193000317,
0.0327763594686985,
0.005308728665113449,
-0.13283874094486237,
-0.0337197445333004,
0.011955255642533302,
0.07070659846067429,
-0.016794554889202118,
-0.020191341638565063,
0.12315282970666885,
0.08944776654243469,
0.08366433531045914,
0.035663194954395294,
0.006626116577535868,
0.03969866782426834,
-0.008333812467753887,
0.0460154227912426,
0.0045291101559996605,
8.211647360056201e-32,
0.052080780267715454,
0.06742069125175476,
0.019641490653157234,
0.009721041657030582,
0.0006642547668889165,
0.03170175105333328,
0.028727389872074127,
0.02548140287399292,
0.05414249375462532,
-0.03125797584652901,
0.10542266815900803,
0.13143572211265564,
-0.08437720686197281,
0.06718683987855911,
-0.006879474967718124,
0.024448033422231674,
0.013384978286921978,
-0.007905377075076103,
0.031424637883901596,
0.03265790641307831,
-0.09401622414588928,
-0.009915724396705627,
-0.03861046954989433,
0.056115251034498215,
-0.06397262215614319,
0.009094112552702427,
-0.11366705596446991,
-0.02857254631817341,
-0.02015013061463833,
-0.05321304500102997,
-0.014990161173045635,
-0.03857996687293053,
0.01496991328895092,
-0.0197453610599041,
0.031077902764081955,
-0.03844829648733139,
-0.037356916815042496,
0.06809400022029877,
-0.012888969853520393,
-0.02410551905632019,
-0.02547544613480568,
-0.030483074486255646,
-0.049160443246364594,
0.07792231440544128,
-0.018726440146565437,
-0.001601851312443614,
0.029907794669270515,
-0.014754507690668106,
0.014933151192963123,
-0.0915425717830658,
-0.04113325476646423,
-0.020214591175317764,
0.0360974557697773,
-0.03401365876197815,
-0.016116604208946228,
-0.07358837872743607,
-0.02191236987709999,
0.03189724683761597,
-0.0012061228044331074,
0.039164457470178604,
-0.04090693220496178,
0.057926759123802185,
-0.0935467928647995,
0.0077132838778197765
] |
68
Chapter 4
4.4 Points of intersection
You can sketch curves of functions to show points of intersection and solutions to equations.
β The x-coordinate(s) at the points of intersection of the curves with equations
y = f(x) and y = g(x) are the solution(s) to the equation f( x) = g( x).
a y
x
CBA
1 3Oy = x(x β 3)
y = x2(1 β x)
b From the graph there are three points
wh
ere the curves cross, labelled A, B
and C. The x -coordinates are given by the
solutions to the equation.
x(x β
3) = x2(1 β x)
x2 β 3 x = x2 β x3
x3 β 3 x = 0
x(x2 β 3) = 0
So
x = 0 or x2 = 3
So x = β ββ―__
3 , 0, ββ―__
3
Sub
stitute into y = x2 (1 β x)
T
he points of intersection are:
A(β ββ―__
3 , 3 + 3 ββ―__
3 )
B(0
, 0)
C( ββ―__
3 , 3 β 3 ββ―__
3 )A cubic curve will eventually get steeper than a
quadratic curve, so the graphs will intersect for some negative value of x.
There are three points of intersection so the equation x(x β 3) = x
2(1 β x) has three real roots.
Multiply out brackets.Collect terms on one side.Factorise.
The graphs intersect for these values of x, so you can substitute into either equation to find the y-coordinates.
Leave your answers in surd form.Example 6
a On the same diagram sk etch the curves with equations y = x(x β 3) and y = x2 (1 β x ).
b Find the coordinates of
the points of intersection.
Example 7
a On the same diagram sk etch the curves with equations y = x2(3x β a) and y = b __ x , where a and b
are positive constants.
b State, gi
ving a reason, the number of real solutions to the equation x2(3x β a) β b __ x = 0
|
[
0.018378593027591705,
0.12505751848220825,
0.02858470380306244,
0.00849381648004055,
-0.038004230707883835,
0.004909105133265257,
0.05547834932804108,
0.031587522476911545,
-0.09973552823066711,
0.015139855444431305,
0.011190874502062798,
-0.060521893203258514,
-0.0472819060087204,
0.02537986822426319,
-0.00779211800545454,
0.029528535902500153,
-0.07752695679664612,
-0.031145378947257996,
0.029695313423871994,
-0.062320418655872345,
-0.05716409161686897,
-0.036488283425569534,
-0.08243821561336517,
-0.11515894532203674,
-0.005612566601485014,
-0.08046092092990875,
-0.019353603944182396,
-0.03556051477789879,
-0.03150998055934906,
-0.010987442918121815,
0.10476371645927429,
0.014902410097420216,
0.0028113853186368942,
-0.019188638776540756,
0.05167228728532791,
0.09342870861291885,
0.06374740600585938,
0.03425324708223343,
0.034737516194581985,
-0.059965163469314575,
-0.035604894161224365,
-0.003183589084073901,
0.018417708575725555,
0.017175739631056786,
0.07332470268011093,
-0.01974678412079811,
0.006545533891767263,
0.03495194390416145,
-0.013480402529239655,
-0.03628746047616005,
-0.017045825719833374,
-0.042257439345121384,
-0.11303368955850601,
0.000536417996045202,
0.0885297954082489,
-0.0012961294269189239,
-0.005851009860634804,
-0.05684906244277954,
0.03630296513438225,
0.12893977761268616,
0.04896804690361023,
0.05096227675676346,
-0.03508500009775162,
0.048154860734939575,
-0.031017759814858437,
0.049114890396595,
0.11441025137901306,
0.021823285147547722,
-0.009829255752265453,
0.11123654991388321,
-0.07445734739303589,
0.016313103958964348,
-0.029491454362869263,
-0.08386710286140442,
0.01805313117802143,
0.012550926767289639,
-0.017439451068639755,
-0.021506808698177338,
-0.0669521614909172,
-0.00010844007192645222,
-0.055119503289461136,
0.015222128480672836,
0.07198939472436905,
0.08933119475841522,
-0.038254182785749435,
0.050999291241168976,
0.0324031338095665,
-0.07242470234632492,
0.04139120131731033,
-0.005414998158812523,
0.003461371874436736,
-0.006824297830462456,
-0.07721185684204102,
-0.028701426461338997,
-0.035621874034404755,
-0.1098385602235794,
0.0023177426774054766,
0.029995063319802284,
0.04092764854431152,
0.0605294406414032,
-0.08659107238054276,
0.02548964135348797,
0.06289483606815338,
0.06713160872459412,
0.03502151370048523,
-0.014462311752140522,
-0.05936829745769501,
-0.028961116448044777,
0.023998994380235672,
-0.07471943646669388,
0.036797940731048584,
-0.012532602064311504,
0.010982688516378403,
0.017655011266469955,
0.0841236263513565,
-0.023348841816186905,
0.07242858409881592,
-0.09594529867172241,
0.055913448333740234,
-0.05842122063040733,
-0.012286407873034477,
-0.022022372111678123,
0.03554293140769005,
-0.02073824219405651,
-0.005899322684854269,
0.023446429520845413,
-0.002103680046275258,
-0.014491312205791473,
-0.012560693547129631,
-0.0639224499464035,
0.024912793189287186,
-0.09809353202581406,
-0.06934449821710587,
0.00480284821242094,
-0.04446195438504219,
-0.04029830917716026,
-0.13953526318073273,
0.06079084053635597,
-0.028366856276988983,
-0.06069841608405113,
0.05667390674352646,
0.047186948359012604,
0.004620022606104612,
0.04912802577018738,
0.034114252775907516,
-0.0008002818212844431,
-0.12112367898225784,
0.019977450370788574,
-0.0048765926621854305,
-0.05217485502362251,
0.007686515338718891,
0.03848278895020485,
0.01649593748152256,
0.1788640171289444,
-0.002608031267300248,
-0.047073185443878174,
0.019490953534841537,
-0.04981542006134987,
-0.0016462128842249513,
-0.009338193573057652,
0.00945178885012865,
0.05066758021712303,
-0.08293348550796509,
0.03923417255282402,
0.039795175194740295,
-0.009912775829434395,
0.03935399278998375,
0.06695782393217087,
0.009976254776120186,
-0.026411382481455803,
-0.035658013075590134,
-0.020879510790109634,
-0.01532580703496933,
0.05331305414438248,
-0.0015878913691267371,
0.015927841886878014,
0.04382825642824173,
0.02348785661160946,
-0.03450429067015648,
-0.009958920069038868,
0.03314347192645073,
-0.05925736576318741,
0.014295210130512714,
-0.03155290335416794,
0.022443683817982674,
-0.009302460588514805,
0.004764549899846315,
-0.005025496706366539,
-0.0001344354241155088,
0.02515978179872036,
-0.05331183969974518,
-0.08537837117910385,
-0.04790986701846123,
0.0923338234424591,
-0.06623763591051102,
-0.026266057044267654,
0.0646536648273468,
0.03378625586628914,
-0.02785477787256241,
0.05459253862500191,
0.0026882700622081757,
-0.026465805247426033,
0.008684436790645123,
-0.055144619196653366,
-0.09029579162597656,
0.020936667919158936,
-0.04752204567193985,
0.05505763366818428,
-0.05295689031481743,
-0.03353498503565788,
-0.006675730459392071,
0.011647619307041168,
-0.08389453589916229,
-0.041307464241981506,
0.04533693566918373,
-0.05838556960225105,
0.005658098962157965,
-0.029718056321144104,
-0.1450309008359909,
0.03793760761618614,
-0.03235534951090813,
0.035858944058418274,
0.033974889665842056,
3.6539424559395e-34,
0.01012422051280737,
-0.011306547559797764,
-0.04668624326586723,
-0.02176865004003048,
-0.025369780138134956,
0.028226615861058235,
0.06962507218122482,
-0.026585346087813377,
0.036156293004751205,
0.05069397762417793,
0.07149907201528549,
-0.017630454152822495,
-0.07858088612556458,
0.04268724471330643,
-0.0001312407257501036,
-0.0343952551484108,
-0.030058523640036583,
-0.075606569647789,
-0.055629629641771317,
-0.014523512683808804,
-0.023267360404133797,
0.009621115401387215,
0.018369298428297043,
-0.051857441663742065,
0.006493499502539635,
-0.02945740707218647,
0.06512273848056793,
-0.07394038885831833,
-0.014860430732369423,
0.061437979340553284,
-0.02270621992647648,
0.004654929507523775,
0.050597596913576126,
0.0006797366077080369,
0.03395542502403259,
0.03711865097284317,
-0.03401878848671913,
-0.03733125701546669,
0.09527510404586792,
-0.10257748514413834,
0.024840429425239563,
-0.0360584519803524,
0.08238060772418976,
0.05700679123401642,
0.02696199156343937,
0.07575920224189758,
-0.030259570106863976,
0.09599292278289795,
0.027445871382951736,
0.005190117284655571,
-0.034705113619565964,
-0.08230087906122208,
0.004679184406995773,
0.038472436368465424,
0.07018712162971497,
0.031956836581230164,
0.06724782288074493,
-0.06396064162254333,
0.02807624824345112,
-0.11342813819646835,
0.04354007914662361,
-0.0380961075425148,
-0.0013555745827034116,
0.04022585228085518,
0.029203234240412712,
-0.07205741852521896,
-0.048873420804739,
-0.0641941949725151,
0.017834996804594994,
0.00912915263324976,
0.023091332986950874,
0.01820506900548935,
-0.02755863405764103,
-0.0666627287864685,
0.01845722086727619,
0.01621941663324833,
-0.034289244562387466,
-0.0013995813205838203,
0.03783635422587395,
-0.09490115195512772,
0.019496778026223183,
0.10211466252803802,
-0.06848039478063583,
0.052816540002822876,
-0.008656238205730915,
0.045427389442920685,
-0.03910977765917778,
0.07284452766180038,
0.11819673329591751,
0.010229921899735928,
0.05833509936928749,
0.056706503033638,
-0.09297950565814972,
0.0018039323622360826,
0.04818425327539444,
7.456386949928303e-32,
-0.07007626444101334,
-0.002064615022391081,
-0.00015151186380535364,
0.008454679511487484,
0.006236620247364044,
-0.018881211057305336,
0.09602814167737961,
-0.004202801268547773,
0.04463842883706093,
-0.013409131206572056,
0.033169668167829514,
0.032549358904361725,
-0.07037948071956635,
0.0014613744569942355,
0.027161112055182457,
0.08383047580718994,
-0.0005038650706410408,
-0.016638727858662605,
0.0533551350235939,
-0.009053133428096771,
-0.08156006783246994,
-0.005486547015607357,
0.032082654535770416,
0.022407356649637222,
0.034054793417453766,
-0.00843673013150692,
0.0011557851685211062,
-0.058727577328681946,
0.01954638585448265,
-0.06459861993789673,
0.015906859189271927,
-0.023844171315431595,
-0.018915073946118355,
0.01164817251265049,
0.13498608767986298,
0.025472814217209816,
-0.1261509358882904,
0.01493625808507204,
0.029912035912275314,
0.014302660711109638,
-0.050282031297683716,
-0.032131973654031754,
-0.03480852022767067,
-0.040557071566581726,
-0.032911818474531174,
-0.08142726868391037,
0.0011212867684662342,
-0.026281552389264107,
0.037823814898729324,
0.00945316907018423,
-0.014130166731774807,
0.04525267332792282,
-0.08117374777793884,
-0.07533315569162369,
0.044201355427503586,
0.011408978141844273,
0.05286458879709244,
-0.021584738045930862,
0.016155906021595,
-0.01448520552366972,
-0.14023493230342865,
0.09636102616786957,
0.0036305587273091078,
0.02050904743373394
] |
69Graphs and transformations
a y
x Oy = x2(3x β a)
ab
xy =b
x
1
3y =
b From the sketch there are only two points
of i
ntersection of the curves. This means
there are only two values of x where
x2 (3x β a) = b __ x
or x2 (3x β a) β b __ x = 0
So this equation has two real solutions.You can sketch curves involving unknown
constants. You should give any points of intersection with the coordinate axes in terms of the constants where appropriate.Problem-solving
Example 8
a Sketch the curves y = 4 ___ x 2 and y = x 2(x β 3) on the same axes.
b Using your sketch, sta
te, with a reason, the number of real solutions to the equation
x 4(x β 3) β 4 = 0.
a y
x
y = x2(x β 3)/four.ss01
x2y =
O 3
b There is a single point of intersection so the
equation x2(x β 3) = 4 ___ x2 has one real solution .
Rearranging:
x 4(x β 3) = 4
x 4(x β 3) β 4 = 0
So this equation has one real solution.Set the functions equal to each other to form an
equation with one real solution, then rearrange the equation into the form given in the question.Problem-solving
You would not be expected to solve this equation in your exam.
1 In each case:
i sketch the two curv
es on the same axes
ii state the number of
points of intersection
iii write down a suitab
le equation which would give the x-coordinates of these points.
(Y ou are not required to solve this equation.)Exercise 4D3x β a = 0 when x = 1 __ 3 a, so the graph of
y = x2(3x β a) touches the x-axis at (0, 0) and
intersects it at ( 1 __ 3 a, 0)
You only need to state the number of solutions.
You donβt need to find the solutions.
|
[
0.008111045695841312,
0.004166868980973959,
-0.04292477294802666,
0.019671732559800148,
0.012028956785798073,
0.008966078981757164,
-0.029686151072382927,
-0.026450615376234055,
-0.02627275511622429,
-0.012827538885176182,
-0.03679974749684334,
-0.06092631816864014,
-0.005618855822831392,
-0.02741081640124321,
-0.01171176414936781,
-0.014346500858664513,
-0.0589560940861702,
-0.018174488097429276,
-0.02630944922566414,
-0.006077247206121683,
-0.016010740771889687,
-0.12207488715648651,
-0.020912395790219307,
-0.10855555534362793,
0.07493521273136139,
-0.04978235065937042,
0.00558669026941061,
0.015880098566412926,
-0.028588440269231796,
-0.014027549885213375,
0.04025309160351753,
0.09431339800357819,
0.028515031561255455,
0.030613599345088005,
0.09069710224866867,
0.05274951457977295,
0.03504078835248947,
0.005910961888730526,
0.046922434121370316,
-0.05426565185189247,
-0.04897310212254524,
0.010322721675038338,
-0.008196428418159485,
-0.03144212067127228,
0.02676231972873211,
-0.06970357894897461,
-0.03904908895492554,
0.05789819359779358,
0.13601157069206238,
-0.021785032004117966,
0.018720868974924088,
-0.03573701158165932,
-0.03769518435001373,
-0.015929458662867546,
0.016464022919535637,
0.02500217594206333,
0.05448097363114357,
0.02393246442079544,
0.014598967507481575,
-0.00046872918028384447,
0.013422209769487381,
0.026811204850673676,
0.042259134352207184,
0.014087177813053131,
0.0016417312435805798,
0.025280628353357315,
0.049036409705877304,
-0.0498131699860096,
-0.021202079951763153,
0.06640167534351349,
-0.05873846635222435,
0.09478477388620377,
-0.028452714905142784,
0.0005642770556733012,
0.033184848725795746,
-0.016375426203012466,
0.00540592335164547,
0.05050787702202797,
-0.015710067003965378,
-0.036400992423295975,
-0.1268494725227356,
-0.053420621901750565,
0.09297604858875275,
0.037139736115932465,
-0.012428103014826775,
0.01771901547908783,
0.04730125144124031,
-0.0045661660842597485,
0.002517752116546035,
0.03964115306735039,
-0.021407345309853554,
0.0769418329000473,
-0.027011321857571602,
-0.11646399646997452,
0.03779832273721695,
-0.1096353828907013,
-0.033234842121601105,
0.010327684693038464,
0.041929565370082855,
0.06737196445465088,
-0.011900672689080238,
0.06245865672826767,
0.039251163601875305,
0.0464835949242115,
-0.029011745005846024,
0.021675987169146538,
-0.023228593170642853,
-0.027865437790751457,
0.0080863693729043,
0.013070947490632534,
-0.07150264084339142,
-0.08696942776441574,
0.05333664268255234,
0.02971928007900715,
0.08607463538646698,
-0.0516260527074337,
0.016130995005369186,
0.034785352647304535,
-0.01956387609243393,
-0.060618143528699875,
0.020484665408730507,
-0.03349481523036957,
0.05771822854876518,
-0.011134185828268528,
-0.09177359938621521,
0.00794240366667509,
-0.08682100474834442,
-0.0004965387634001672,
-0.022304780781269073,
0.000677662726957351,
0.05129192769527435,
-0.072512686252594,
-0.06260078400373459,
-0.007441012188792229,
-0.03466411307454109,
0.016210848465561867,
-0.1135726273059845,
0.09559057652950287,
-0.012164338491857052,
-0.04149773344397545,
0.022705238312482834,
0.1419827938079834,
0.07646912336349487,
0.04993031546473503,
-0.0404876172542572,
-0.014282837510108948,
-0.05704312026500702,
0.041608963161706924,
-0.03994809463620186,
-0.02465479075908661,
-0.03663637116551399,
0.014822251163423061,
0.012565082870423794,
0.15140792727470398,
0.011848066933453083,
-0.04403936490416527,
-0.01729712262749672,
-0.06084282323718071,
-0.10400660336017609,
0.018985318019986153,
0.009080380201339722,
0.0016970045398920774,
-0.020417246967554092,
0.052816011011600494,
-0.0047383117489516735,
0.005323980003595352,
0.0907401293516159,
0.09526094049215317,
-0.024487340822815895,
0.03204985707998276,
-0.0011346094543114305,
0.02837533876299858,
-0.0025946919340640306,
0.09313396364450455,
0.024695493280887604,
0.06009212136268616,
0.05839302018284798,
-0.025544047355651855,
-0.06921190023422241,
0.02175016514956951,
0.026376847177743912,
-0.057135630398988724,
-0.029570743441581726,
0.03858742490410805,
-0.007225071080029011,
-0.011759716086089611,
0.04600859060883522,
-0.008260658010840416,
-0.02031164988875389,
0.004731175024062395,
-0.014767788350582123,
-0.022899247705936432,
-0.04471883922815323,
0.04236578196287155,
-0.1044682115316391,
-0.025587746873497963,
0.05239453166723251,
-0.022393397986888885,
-0.10060545802116394,
0.019694533199071884,
0.06476906687021255,
0.010468602180480957,
-0.0021442663855850697,
0.01145558338612318,
0.03604179248213768,
0.02334088459610939,
-0.042000044137239456,
0.01609223522245884,
-0.06358905136585236,
0.03775370866060257,
-0.03949640318751335,
0.001965709263458848,
-0.008622074499726295,
0.03691486269235611,
-0.01672208309173584,
-0.07717370986938477,
0.03686514124274254,
-0.014221803285181522,
-0.05103960260748863,
0.01993071474134922,
-0.10021551698446274,
-0.058799318969249725,
0.10757172852754593,
1.1472982821078918e-33,
0.05971703678369522,
0.018690211698412895,
-0.11242391169071198,
-0.04354158788919449,
0.01336120255291462,
-0.056157179176807404,
0.08931413292884827,
-0.033993907272815704,
0.05666014552116394,
-0.03077598661184311,
0.0976349487900734,
0.02867945469915867,
-0.011610988527536392,
-0.022732321172952652,
0.0020837828051298857,
-0.07378212362527847,
0.016924329102039337,
-0.07091707736253738,
0.010931714437901974,
-0.09427560120820999,
0.017586268484592438,
-0.001196324243210256,
-0.0014274261193349957,
0.009471730329096317,
0.006619640626013279,
0.056396517902612686,
-0.01031856331974268,
-0.05098259076476097,
-0.06725513935089111,
0.05968879163265228,
-0.09106823801994324,
-0.030451228842139244,
0.02725815400481224,
0.026680007576942444,
-0.03694368898868561,
0.01118298526853323,
-0.06415344774723053,
-0.0035909144207835197,
0.10861066728830338,
-0.027474556118249893,
-0.01916365884244442,
0.03174566477537155,
0.07312988489866257,
0.0470719113945961,
-0.025156594812870026,
0.12142080813646317,
0.04501435533165932,
0.06003734841942787,
-0.018190057948231697,
-0.03134353458881378,
-0.031608279794454575,
-0.12499124556779861,
0.015714913606643677,
-0.05002913624048233,
0.12375176697969437,
0.013620197772979736,
0.003199968719854951,
-0.052987076342105865,
0.008420371450483799,
-0.04191594198346138,
0.0019804961048066616,
-0.04022545367479324,
-0.00493339030072093,
0.03711162880063057,
-0.052611470222473145,
-0.02242175303399563,
-0.06277012079954147,
-0.1223529577255249,
-0.03532668575644493,
-0.039812974631786346,
0.017747247591614723,
0.061979249119758606,
-0.08209795504808426,
-0.06062248721718788,
0.05907442793250084,
-0.046468302607536316,
-0.007124222815036774,
0.030834008008241653,
0.06805524975061417,
-0.034510962665081024,
-0.06815126538276672,
-0.001672455808147788,
-0.029307784512639046,
0.03844573348760605,
0.03220073878765106,
0.007496816571801901,
0.066790372133255,
0.07335265725851059,
0.0849844366312027,
0.05808214098215103,
0.1127796396613121,
0.02526642195880413,
-0.13157442212104797,
0.007397052366286516,
0.10176821798086166,
6.585650089005851e-32,
-0.012480616569519043,
0.09877559542655945,
0.018604086712002754,
-0.02528022602200508,
-0.024579625576734543,
0.010033528320491314,
0.0017086752923205495,
-0.025283221155405045,
0.02171679399907589,
-0.054149337112903595,
0.02645402029156685,
0.07996000349521637,
-0.06542517244815826,
0.07422231882810593,
0.0033993951510638,
0.022251160815358162,
0.005388846155256033,
0.012015195563435555,
0.05715714395046234,
-0.05480686202645302,
-0.1298791915178299,
0.04269915819168091,
0.04255947470664978,
0.06420467793941498,
-0.050375692546367645,
-0.05139460414648056,
-0.05108991637825966,
-0.06907342374324799,
-0.07692161202430725,
-0.017654314637184143,
0.014025491662323475,
-0.0030229564290493727,
-0.013908831402659416,
-0.04796812683343887,
0.10299232602119446,
0.01022004708647728,
-0.06779283285140991,
0.06385430693626404,
0.03519845008850098,
-0.04495710879564285,
-0.0073973676189780235,
0.0048337820917367935,
-0.02175378054380417,
0.03216046467423439,
0.005025457125157118,
-0.027298569679260254,
0.008380360901355743,
-0.03723941370844841,
0.06476587057113647,
-0.037331946194171906,
0.005669084843248129,
-0.004053172655403614,
0.04923803731799126,
-0.0030676675960421562,
0.008961062878370285,
-0.06434256583452225,
-0.02508123405277729,
-0.009753447957336903,
-0.04945699870586395,
0.007121618837118149,
-0.05713803693652153,
0.07132501900196075,
-0.10584506392478943,
-0.002469941508024931
] |
70
Chapter 4
a y =
x2, y = x(x2 β 1) b y = x(x + 2), y = β 3 __ x c y = x2, y = (x + 1)(x β 1)2
d y = x2(1 β x), y = β 2 __ x e y = x(x β 4), y = 1 __ x f y = x(x β 4), y = β 1 __ x
g y =
x(x β 4), y = (x β 2)3 h y = βx3, y = β 2 __ x i y = βx3, y = x2
j y = β x3, y = βx(x + 2) k y = 4, y = x(x β 1)(x + 2)2 l y = x3, y = x2(x + 1)2
2 a On the same axes sketch the curv es given by y = x2(x β 3) and y = 2 __ x
b Explain how your sk
etch shows that there are only two real solutions to the equation
x 3(x β 3) = 2.
3 a On the same axes sketch the curv
es given by y = (x + 1)3 and y = 3x(x β 1).
b Explain how your sk
etch shows that there is only one real solution to the equation
x 3 + 6x + 1 = 0.
4 a On the same axes sketch the curv
es given by y = 1 __ x and y = βx(x β 1)2.
b Explain how your sk
etch shows that there are no real solutions to the equation
1 + x2(x β 1)2 = 0.
5 a On the same axes sketch the curv
es given by y = x2(x + a)
and y = b __ x where a and b are both positive
constants. (5 marks)
b Using your sketch, sta
te, giving a reason, the number of
real solutions to the equation x 4 + ax3 β b = 0. (1 mark)
6 a On the same set of axes sk
etch the graphs of
y =
4 __ x2 and y = 3x + 7. (3 marks)
b Write down the n
umber of real solutions to the equation 4 __ x2 = 3x + 7. (1 mark)
c Show that y
ou can rearrange the equation to give (x + 1)(x + 2)(3x β 2) = 0. (2 marks)
d Hence determine the exact coor
dinates of the points of intersection. (3 marks)
7 a On the same axes sketch the curv
e y = x3 β 3x2 β 4x and the line y = 6x.
b Find the coordinates of
the points of intersection.
8 a On the same axes sketch the curv
e y = (x2 β 1)(x β 2) and the line y = 14x + 2.
b Find the coordinates of
the points of intersection.
9 a On the same axes sketch the curv
es with equations y = (x β 2)(x + 2)2 and y = βx2 β 8.
b Find the coordinates of
the points of intersection.
10 a Sketch the gra
phs of y = x2 + 1 and 2y = x β 1. (3 marks)
b Explain why ther
e are no real solutions to the equation 2x2 β x + 3 = 0. (2 marks)
c Work out the r
ange of values of a such that the graphs of y = x2 + a and 2y = x β 1
have two points of intersection. (5 marks)E/P
Even though you donβt know
the values of a and b , you
know they are positive, so you know the shapes of the graphs. You can label the point a on the x -axis on your
sketch of
y = x2(x + a).Problem-solving
E
P
P
E/P
|
[
0.006487688515335321,
0.04751107469201088,
-0.014701266773045063,
-0.07448455691337585,
-0.022692671045660973,
0.0914299488067627,
0.05339983105659485,
0.021131252869963646,
-0.08924069255590439,
0.024006370455026627,
0.05221424996852875,
-0.03407466039061546,
0.009938026778399944,
-0.034749582409858704,
-0.09107568114995956,
-0.012044643051922321,
-0.10545535385608673,
-0.025199105963110924,
-0.009287030436098576,
-0.02703288570046425,
0.041956640779972076,
-0.042883969843387604,
-0.050047438591718674,
-0.03138474002480507,
0.06971358507871628,
0.0069323210045695305,
-0.029267149046063423,
-0.037782084196805954,
-0.039910174906253815,
-0.07444414496421814,
-0.03534660115838051,
0.031943630427122116,
0.06006123125553131,
-0.012323418632149696,
0.06452732533216476,
0.11090271919965744,
0.08249203115701675,
0.047775521874427795,
0.020473487675189972,
-0.02642926573753357,
-0.06515186280012131,
-0.019537825137376785,
0.007408004719763994,
0.017534727230668068,
0.028180792927742004,
-0.018936995416879654,
-0.05264173075556755,
-0.010646125301718712,
-0.011427651159465313,
-0.05978890508413315,
-0.00016691192286089063,
0.01606777496635914,
-0.05958086997270584,
0.05849500745534897,
0.028993209823966026,
-0.07527520507574081,
-0.005975321866571903,
0.02025974541902542,
-0.0038662420120090246,
0.059655915945768356,
-0.037567514926195145,
-0.008022363297641277,
-0.0004183245764579624,
0.010119188576936722,
-0.02375263348221779,
0.0803699716925621,
0.0016558022471144795,
-0.09414009004831314,
-0.02310926839709282,
0.11156146973371506,
-0.12692616879940033,
0.020739402621984482,
-0.06330857425928116,
-0.054169122129678726,
0.025711214169859886,
-0.003695616265758872,
-0.013297045603394508,
0.030064759775996208,
-0.00988525990396738,
-0.03696266561746597,
-0.0670807883143425,
-0.012086049653589725,
0.13317464292049408,
0.018056655302643776,
-0.005568955093622208,
-0.007432733196765184,
0.05360513925552368,
-0.023760518059134483,
0.04293148219585419,
-0.03805050626397133,
-0.010241848416626453,
-0.04779323190450668,
-0.02330033667385578,
-0.013196192681789398,
-0.011054180562496185,
-0.09315718710422516,
0.03462694585323334,
-0.06828827410936356,
0.07550704479217529,
0.12061626464128494,
-0.021184707060456276,
0.019918110221624374,
0.041688863188028336,
0.03544081375002861,
0.01746929995715618,
-0.012501271441578865,
0.038695622235536575,
-0.04002152383327484,
-0.0018175175646319985,
-0.11836062371730804,
-0.01863480545580387,
-0.04548842832446098,
-0.0069265710189938545,
0.02796107530593872,
0.00733683817088604,
-0.11211124807596207,
0.007941915653645992,
-0.08614110201597214,
-0.04135425388813019,
-0.03692321479320526,
0.034892115741968155,
-0.04112647473812103,
0.11807776242494583,
0.05288436636328697,
-0.009933396242558956,
0.024318182840943336,
0.008487436920404434,
-0.003395536681637168,
-0.03208170458674431,
-0.10495982319116592,
0.03134581446647644,
-0.041192252188920975,
-0.05428731068968773,
-0.035975851118564606,
-0.03080521710216999,
-0.015170203521847725,
-0.03532613441348076,
0.05998501554131508,
-0.05607777461409569,
-0.010298572480678558,
-0.08876027911901474,
0.09210748225450516,
0.027873266488313675,
0.0016877760645002127,
0.00856687780469656,
-0.021544059738516808,
-0.05070672929286957,
-0.07224022597074509,
0.05169183760881424,
0.1223287582397461,
0.008261810056865215,
-0.015223568305373192,
0.005373833235353231,
0.10540767014026642,
-0.028084680438041687,
-0.05607861652970314,
-0.04552461579442024,
-0.01570775732398033,
0.027228785678744316,
-0.05020880699157715,
-0.03445299342274666,
0.04840314760804176,
-0.01299505028873682,
0.08831942081451416,
0.04327087849378586,
0.00287385331466794,
-0.0013052471913397312,
0.08052975684404373,
-0.024570375680923462,
-0.014995578676462173,
0.0023607334587723017,
-0.05450621619820595,
0.07921303808689117,
0.08258076012134552,
0.017104625701904297,
0.0340869314968586,
0.12174476683139801,
0.04601072892546654,
-0.051156770437955856,
0.05799923837184906,
0.004127363208681345,
-0.06051531806588173,
-0.020992746576666832,
0.010236666537821293,
-0.02265472151339054,
-0.0173464547842741,
-0.10258717834949493,
0.10275646299123764,
-0.005860466975718737,
0.04209320992231369,
-0.0054898615926504135,
-0.031975459307432175,
0.052017588168382645,
0.08760741353034973,
-0.1405317336320877,
-0.0529882088303566,
-0.007794107776135206,
0.009380258619785309,
-0.10517861694097519,
0.006923574022948742,
0.059617068618535995,
0.028864428400993347,
0.08040542900562286,
0.02260485291481018,
-0.030043121427297592,
-0.08744018524885178,
-0.09922393411397934,
-0.06461358815431595,
-0.04822386056184769,
-0.0007727502379566431,
-0.016240978613495827,
0.07822734862565994,
-0.09099815040826797,
-0.039484020322561264,
-0.014212099835276604,
-0.10984349250793457,
0.06821493804454803,
-0.041579458862543106,
-0.04590868204832077,
-0.007930011488497257,
-0.012296659871935844,
-0.048315536230802536,
-0.025532882660627365,
1.1610176787144302e-33,
-0.04086841642856598,
0.039129871875047684,
-0.08972170203924179,
-0.0439961813390255,
0.009299643337726593,
-0.04586225003004074,
0.04714826121926308,
-0.06457746028900146,
0.068409264087677,
0.07169459015130997,
0.03720194846391678,
0.004414120223373175,
-0.07366576045751572,
-0.007932246662676334,
-0.016773102805018425,
-0.03196141496300697,
-0.07228774577379227,
0.055322498083114624,
0.030031617730855942,
0.010842629708349705,
-0.051159925758838654,
-0.01724875718355179,
-0.024467911571264267,
-0.0353880450129509,
-0.030529512092471123,
0.017542731016874313,
0.06995600461959839,
-0.06535235792398453,
-0.006656994111835957,
-0.024556441232562065,
-0.01812107115983963,
-0.06755564361810684,
0.07315218448638916,
0.0733238235116005,
-0.007149310316890478,
-0.04385961592197418,
0.027160312980413437,
-0.0266273096203804,
-0.055528681725263596,
0.012157673947513103,
0.07634086161851883,
0.030223065987229347,
0.01175228413194418,
0.03377727046608925,
0.03708471357822418,
0.018892377614974976,
0.03208102658390999,
0.04536458104848862,
0.015404129400849342,
0.04338204115629196,
-0.04092470183968544,
-0.05737154185771942,
-0.031047536060214043,
0.04870986193418503,
0.06787358969449997,
0.019809074699878693,
0.0204605869948864,
-0.07581061869859695,
0.04893786460161209,
-0.05973871797323227,
-0.0007831219118088484,
-0.00947173498570919,
0.04395624250173569,
0.039561040699481964,
0.006572597194463015,
-0.09356865286827087,
-0.08933109790086746,
-0.0012748248409479856,
-0.009936753660440445,
-0.023212170228362083,
-0.0014422948006540537,
0.09754937142133713,
-0.029016394168138504,
-0.03386582434177399,
-0.02229982614517212,
-0.0055373297072947025,
-0.02089916355907917,
0.000542341498658061,
-0.019513610750436783,
-0.0202238317579031,
-0.08208715170621872,
0.002172179752960801,
0.01812976598739624,
0.011301509104669094,
-0.004773813299834728,
0.0012060068547725677,
0.13155962526798248,
0.10440731793642044,
0.1111086905002594,
0.03317786008119583,
0.03052639029920101,
-0.028999775648117065,
0.044004395604133606,
-0.012360841035842896,
0.004486164078116417,
1.0618798830794364e-31,
-0.009652075357735157,
-0.0395633764564991,
-0.0420408733189106,
0.05983839929103851,
0.029770424589514732,
0.031904395669698715,
0.021418415009975433,
-0.0036621522158384323,
0.0068473536521196365,
-0.03375192731618881,
0.09199728071689606,
0.1329914927482605,
-0.07737759500741959,
0.06575050950050354,
-0.05480123311281204,
0.03413677215576172,
-0.07938722521066666,
-0.03212679922580719,
-0.051814932376146317,
-0.016668230295181274,
-0.03366035968065262,
-0.012954764999449253,
-0.08685263991355896,
0.012772015295922756,
0.08066510409116745,
0.01711459830403328,
-0.06452727317810059,
-0.04180607944726944,
0.04067547246813774,
0.014210180379450321,
0.022389670833945274,
-0.012848378159105778,
0.08957932144403458,
-0.09991902858018875,
0.040714081376791,
0.029988925904035568,
-0.03139827772974968,
0.03161370009183884,
0.006005047354847193,
-0.004611331503838301,
0.005810614675283432,
-0.007660359144210815,
-0.004007458686828613,
0.02062671259045601,
0.02948959544301033,
-0.03729360178112984,
0.020724734291434288,
-0.046920500695705414,
0.009084843099117279,
-0.026009568944573402,
-0.07106553018093109,
-0.016208728775382042,
0.04196780174970627,
-0.004076993092894554,
0.08697137981653214,
-0.046984925866127014,
-0.021650968119502068,
0.0017736998852342367,
-0.008970704860985279,
-0.024139972403645515,
-0.02095869928598404,
0.106825090944767,
-0.04703768342733383,
0.056719254702329636
] |
71Graphs and transformations
11 a Sketch the gra
phs of y = x2(x β 1)(x + 1) and y = 1 _ 3 x3 + 1. (5 marks)
b Find the number of r
eal solutions to the equation 3x2(x β 1)(x + 1) = x3 + 3. (1 mark)E/P
4.5 Translating graphs
You can transform the graph of a function by altering the function. Adding or subtracting a constant
βoutsideβ the function translates a graph vertically.
β The graph of y = f(x) + a is a translation of the graph y = f(x) by the vector ( 0 a ) .
Adding or subtracting a constant βinsideβ the function translates the graph horizontally.
β The graph of y = f(x + a) is a translation of the graph y = f(x) by the vector ( βa 0 ) .
y
x O2 4 1 3 β2 β4β5 β1 β3123456
β1y = f(x + 2) is a translation ( β2 0 ) , or 2 units in the
direction of the negative x-axis.y = f(x) + 1 is a translation ( 0 1 ) , or 1 unit in the
direction of the positive y-axis.
Example 9
Sketch the graphs of:
a y =
x2 b y = (x β 2)2 c y = x2 + 2
a y
x O
b y
y = (x β 2)2
x/four.ss01
2 OThis is a translation by vector ( 2 0 ) .
Remember to mark on the intersections with the
axes.
|
[
-0.0686345025897026,
0.00405403645709157,
-0.0477261021733284,
-0.047871969640254974,
-0.050703033804893494,
0.014110229909420013,
-0.005732616409659386,
-0.04154257848858833,
0.011037252843379974,
0.02051602117717266,
0.018284481018781662,
-0.02727763168513775,
-0.005836542695760727,
0.009617340750992298,
-0.011987634934484959,
-0.07238684594631195,
-0.010002721101045609,
0.014293796382844448,
-0.04770864546298981,
-0.0812118798494339,
-0.015060468576848507,
-0.08941169083118439,
-0.10817117989063263,
-0.06615594029426575,
0.131564199924469,
-0.045805320143699646,
-0.025439107790589333,
-0.04440870136022568,
-0.06231372803449631,
-0.06762129813432693,
0.008161941543221474,
0.0642092376947403,
0.02159075252711773,
-0.039071258157491684,
0.07764340937137604,
0.0016005000798031688,
0.015191280283033848,
-0.00045625591883435845,
0.008063297718763351,
0.013032229617238045,
-0.012433765456080437,
0.08269534260034561,
-0.017686642706394196,
-0.02661590278148651,
0.10827937722206116,
-0.03186091035604477,
-0.03677717596292496,
-0.009525532834231853,
0.03299064189195633,
-0.04523563012480736,
0.026076912879943848,
0.0570216104388237,
-0.051753364503383636,
0.0455949567258358,
0.025459522381424904,
-0.016095789149403572,
0.0607011653482914,
0.010201522149145603,
-0.01792922057211399,
0.06668267399072647,
-0.01853473111987114,
0.04843716323375702,
-0.02151518501341343,
0.04962484911084175,
-0.004453373607248068,
0.040099386125802994,
0.004591764882206917,
-0.06970258802175522,
-0.011392735876142979,
0.10222731530666351,
-0.10285741090774536,
0.040521133691072464,
-0.04458179697394371,
-0.0904177576303482,
0.03286183252930641,
0.02942027524113655,
0.01094019040465355,
-0.005534206982702017,
-0.0655035525560379,
-0.05923400819301605,
-0.043077655136585236,
0.018920283764600754,
0.06974364817142487,
0.06026756018400192,
-0.0025289461482316256,
0.04025755822658539,
0.025335010141134262,
0.005137946456670761,
-0.03775406628847122,
0.006800505332648754,
-0.006777243223041296,
0.08488473296165466,
-0.00820108875632286,
-0.025568662211298943,
0.0027042056899517775,
-0.08107801526784897,
0.017537707462906837,
-0.0234712865203619,
0.052795469760894775,
0.0721379816532135,
-0.0642227977514267,
0.07549258321523666,
0.03731173649430275,
0.011727388948202133,
-0.0514807365834713,
-0.06179993972182274,
0.04356386139988899,
-0.025590822100639343,
0.03526540473103523,
-0.026275943964719772,
-0.01842667907476425,
-0.058995768427848816,
0.009665883146226406,
0.1045306846499443,
0.04738523066043854,
-0.053613245487213135,
0.026247410103678703,
-0.006851101294159889,
0.0344858281314373,
-0.05476919561624527,
0.03716667741537094,
-0.04345359653234482,
0.02399379201233387,
-0.0262918584048748,
-0.07308807969093323,
-0.014537016861140728,
-0.06060992181301117,
0.0005495853256434202,
0.03534941002726555,
0.02069968357682228,
0.07505165785551071,
-0.031682442873716354,
0.009360759519040585,
-0.04661150276660919,
-0.07405098527669907,
-0.03228684142231941,
-0.04822726175189018,
0.01830689236521721,
-0.023979533463716507,
0.09431447088718414,
0.058914486318826675,
0.10730675607919693,
-0.0017462638206779957,
0.05369257554411888,
0.04311997443437576,
-0.02566215582191944,
-0.041246447712183,
0.04172858968377113,
-0.04947633668780327,
0.00041144402348436415,
-0.06434962153434753,
0.06400424987077713,
0.003873229492455721,
0.037160370498895645,
0.03210059180855751,
-0.05672839283943176,
-0.04436672851443291,
-0.06334017962217331,
-0.05446800962090492,
0.03287892043590546,
-0.09621977806091309,
0.00962953083217144,
-0.04301733896136284,
0.06359443068504333,
0.05285046622157097,
-0.01464688591659069,
0.08664168417453766,
0.0941484197974205,
-0.015356160700321198,
0.05049007385969162,
0.015571731142699718,
-0.0235621128231287,
0.012192082591354847,
0.09051479399204254,
0.016682375222444534,
-0.01880936697125435,
0.1007159873843193,
0.03124055452644825,
-0.05337485671043396,
0.05063927546143532,
0.0699426457285881,
-0.04051358252763748,
0.01408930029720068,
0.03524725139141083,
-0.08129649609327316,
-0.07287582010030746,
0.0007435693987645209,
0.09418582171201706,
-0.027560973539948463,
0.0322740264236927,
-0.031053951010107994,
-0.0006502672913484275,
-0.011783326976001263,
0.12905345857143402,
-0.09236381202936172,
-0.07272005826234818,
-0.0072722905315458775,
0.03181685507297516,
-0.08465531468391418,
-0.017828382551670074,
0.012024862691760063,
0.0488416887819767,
0.11727536469697952,
0.011391790583729744,
-0.042875997722148895,
-0.014497259631752968,
-0.06798230111598969,
-0.041854992508888245,
-0.09380143880844116,
-0.11205990612506866,
0.0018892004154622555,
0.03489173576235771,
-0.0048230611719191074,
-0.029565472155809402,
-0.03461305424571037,
-0.05797162652015686,
0.015853293240070343,
-0.039866626262664795,
-0.09266886860132217,
0.030022546648979187,
0.04466727748513222,
-0.07451190054416656,
0.02899770252406597,
-4.808763718221662e-34,
-0.0052154758013784885,
0.057045694440603256,
-0.060188084840774536,
-0.011755402199923992,
-0.04076429829001427,
-0.08742590993642807,
0.06974641233682632,
0.07043293863534927,
0.09493940323591232,
-0.00998859852552414,
0.0882057324051857,
0.05009735748171806,
-0.0434892475605011,
0.005043966695666313,
0.01799403689801693,
-0.06973687559366226,
0.02138250693678856,
0.015630481764674187,
-0.0993179902434349,
0.0074716187082231045,
0.03826539218425751,
0.023851510137319565,
0.006263820454478264,
0.02849389612674713,
-0.04596162587404251,
0.04807429760694504,
0.01830027811229229,
-0.08962682634592056,
0.002057464327663183,
-0.031981926411390305,
-0.07520569860935211,
-0.009920492768287659,
-0.004165849182754755,
0.008259220980107784,
-0.04715566337108612,
0.007489620242267847,
-0.003355064196512103,
-0.054905254393815994,
0.00027645882801152766,
0.07425443083047867,
0.03528576344251633,
-0.007863864302635193,
0.052379805594682693,
-0.005111068021506071,
-0.018804755061864853,
0.026400024071335793,
-0.0009697367204353213,
0.06424636393785477,
-0.04865075647830963,
-0.005511990748345852,
-0.048233967274427414,
-0.056057192385196686,
-0.03485991805791855,
0.03735059127211571,
0.1615992933511734,
0.04822932183742523,
0.009113054722547531,
0.020259004086256027,
-0.031227298080921173,
-0.11909592151641846,
-0.0026830032002180815,
-0.03693192079663277,
-0.05194015055894852,
0.004292103927582502,
-0.09605292975902557,
-0.08180471509695053,
-0.07345293462276459,
-0.03784910961985588,
-0.01064123772084713,
-0.007508900482207537,
0.12436230480670929,
0.014079255051910877,
0.05190019682049751,
-0.04621235281229019,
-0.009776460006833076,
-0.06189970299601555,
-0.06387798488140106,
-0.002798332367092371,
-0.009779665619134903,
-0.10460752993822098,
-0.024169309064745903,
-0.03591408580541611,
-0.03950844332575798,
0.08304339647293091,
0.02074158936738968,
-0.03825182095170021,
0.05223654583096504,
0.08898510783910751,
0.07515303045511246,
0.0027667726390063763,
0.04056117311120033,
0.011270469054579735,
-0.07769067585468292,
0.014538089744746685,
0.03624328225851059,
7.734953250197467e-32,
0.03601158410310745,
0.004718104377388954,
-0.019815737381577492,
0.04503854736685753,
0.023855337873101234,
-0.003555034287273884,
0.08170538395643234,
-0.04523807391524315,
-0.06606503576040268,
-0.06102157011628151,
0.019572114571928978,
0.06968140602111816,
-0.018874363973736763,
0.04389932006597519,
-0.07296017557382584,
-0.032927900552749634,
-0.0010121524101123214,
0.03858988359570503,
0.018455103039741516,
0.018552672117948532,
-0.08123032003641129,
0.038274917751550674,
-0.041077904403209686,
0.05024482309818268,
-0.04122927412390709,
-0.04445536062121391,
-0.02077881619334221,
-0.054925523698329926,
-0.002689244458451867,
0.0034476458095014095,
0.004501380957663059,
-0.010039827786386013,
0.021164579316973686,
-0.07576769590377808,
0.12519091367721558,
-0.000855760183185339,
-0.0964512974023819,
0.040888141840696335,
-0.02582433447241783,
-0.021114198490977287,
-0.05474865064024925,
-0.0013517627958208323,
-0.03289405629038811,
0.01999483071267605,
-0.022788578644394875,
0.05761513486504555,
0.020306680351495743,
-0.06517130136489868,
0.09599707275629044,
-0.025102755054831505,
-0.025206828489899635,
0.04553212970495224,
0.08137047290802002,
-0.07161732763051987,
0.005240228492766619,
0.02404153160750866,
-0.01574140414595604,
-0.03415518254041672,
-0.006110052112489939,
0.07290726155042648,
0.03373696655035019,
0.1189960464835167,
0.046868450939655304,
0.030167223885655403
] |
72
Chapter 4
c y = x2 + 2
y
x2
OThis is a translation by vector ( 0 2 ) .
Remember to mark on the y-axis intersection.
Example 10
f(x) = x3
g(x) = x(x β 2)
Sketch the following graphs, indicating any points where the curves cross the axes:a
y =
f(x + 1)
b y =
g(x + 1)
a The graph of f( x) is
y
x Oy = f( x) = x3
So the graph of y = f( x + 1) is
y
x Oy = f( x + 1) = ( x + 1)3
β11
b g(x) =
x(x β 2)
The curve is y = x (x β 2)
0 = x(x β 2)
So x = 0 or x = 2
y
xy = g( x) = x(x β 2)
2 OFirst sketch y = f(x).
This is a translation of the graph of y = f(x) by
vector ( β1 0 ) .
You could also write out the equation as
y = (x + 1)3 and sketch the graph directly.
First sketch g(x).Put y = 0 to find where the curve crosses the x-axis. Explore translations of the
gr
aph of y = x3 using GeoGebra.Online
|
[
-0.030355287715792656,
0.012230916880071163,
-0.0545712448656559,
-0.08790231496095657,
-0.049356989562511444,
0.060417257249355316,
0.03651876747608185,
-0.018507512286305428,
-0.052644215524196625,
0.09456232935190201,
-0.00821249932050705,
-0.04320719838142395,
0.006469930522143841,
0.033455926924943924,
-0.06415371596813202,
0.011098621413111687,
-0.11463729292154312,
0.01324777863919735,
0.009784810245037079,
-0.12836873531341553,
0.013216754421591759,
-0.014872491359710693,
-0.09083743393421173,
-0.10324122756719589,
0.04052391275763512,
-0.03804616257548332,
-0.009523691609501839,
-0.06493716686964035,
-0.03570474311709404,
-0.035818833857774734,
0.018733225762844086,
0.018167344853281975,
-0.06588742882013321,
0.0013854437274858356,
0.023117544129490852,
0.01986023783683777,
0.011788062751293182,
-0.016568318009376526,
0.0687343031167984,
-0.004629557486623526,
-0.08066507428884506,
0.006815788801759481,
-0.012987462803721428,
-0.0007127904100343585,
0.10054191946983337,
0.012293164618313313,
-0.04921701177954674,
-0.047196295112371445,
0.0035632008221000433,
-0.023290501907467842,
0.002870290307328105,
0.025801202282309532,
-0.06664988398551941,
-0.00013212757767178118,
0.032893210649490356,
0.0257563516497612,
0.010198465548455715,
-0.030423887073993683,
0.04234788194298744,
0.041550133377313614,
0.03140778839588165,
0.008951240219175816,
0.021279804408550262,
0.04658579081296921,
0.024852978065609932,
0.05447617545723915,
-0.003884260542690754,
0.0084242457523942,
-0.03187137097120285,
0.10683202743530273,
-0.13640619814395905,
0.05058157816529274,
-0.0008013257756829262,
-0.02180028147995472,
0.00849812664091587,
-0.06599067896604538,
0.02477683126926422,
-0.012244480662047863,
0.05005590245127678,
-0.0015299717197194695,
-0.05426357686519623,
0.029504919424653053,
0.06985016167163849,
0.03456168994307518,
-0.03920966386795044,
-0.012007628567516804,
0.01097628939896822,
-0.0740622729063034,
0.027518460527062416,
-0.03888773173093796,
0.013757247477769852,
0.014550597406923771,
-0.004744495265185833,
-0.023491786792874336,
0.07328660041093826,
-0.10640324652194977,
0.08163096010684967,
-0.05748879909515381,
0.04280555620789528,
0.09314316511154175,
-0.024521082639694214,
0.043418657034635544,
0.03491426259279251,
0.0660528615117073,
0.043831389397382736,
-0.009363214485347271,
0.04191075265407562,
-0.040427129715681076,
-0.0193626768887043,
-0.053207628428936005,
-0.023760071024298668,
-0.033921170979738235,
-0.05000801011919975,
0.08703979104757309,
0.10503382980823517,
-0.07041268050670624,
0.0698641911149025,
-0.07579363882541656,
0.021255308762192726,
-0.04926694929599762,
0.05119611695408821,
-0.0494399294257164,
-0.014224390499293804,
0.003267691470682621,
-0.05611111968755722,
-0.058242421597242355,
-0.0025964484084397554,
-0.021000975742936134,
0.03807689622044563,
-0.053199414163827896,
0.018291296437382698,
-0.054665546864271164,
-0.028673656284809113,
-0.058770060539245605,
-0.017065517604351044,
0.044297248125076294,
-0.10627766698598862,
0.05530455335974693,
-0.030749546363949776,
-0.03037181869149208,
0.057515162974596024,
0.11099524796009064,
0.06955613940954208,
-0.0038888256531208754,
0.002705635968595743,
-0.03214515745639801,
-0.06375473737716675,
0.04274076223373413,
0.04926351457834244,
0.06137979403138161,
-0.05951967462897301,
0.010778450407087803,
-0.003799774218350649,
0.10513059049844742,
0.007821577601134777,
-0.03969958797097206,
0.006894766818732023,
-0.07960744202136993,
0.017780892550945282,
-0.07459445297718048,
-0.03391686826944351,
-0.010673802345991135,
-0.06219571828842163,
0.00787925161421299,
0.04634038731455803,
-0.016083303838968277,
-0.04967574402689934,
0.13981345295906067,
0.001770984148606658,
-0.0019268819596618414,
-0.00012918496213387698,
-0.037639714777469635,
0.002924651838839054,
0.07530466467142105,
0.03798893094062805,
-0.051535382866859436,
0.08737362176179886,
0.01264127530157566,
-0.04545825719833374,
0.019222918897867203,
0.06142761558294296,
-0.05273248255252838,
-0.011411608196794987,
-0.004634592682123184,
-0.040564607828855515,
-0.024900782853364944,
-0.03413751721382141,
0.03677641600370407,
-0.044539645314216614,
0.059553083032369614,
-0.005708097945898771,
-0.018788550049066544,
-0.029824374243617058,
0.11217375099658966,
-0.11833274364471436,
-0.0242001935839653,
0.05939820781350136,
0.08065565675497055,
-0.025838453322649002,
0.020114103332161903,
-0.03903632238507271,
0.10099417716264725,
0.03201545402407646,
0.008453495800495148,
-0.014198497869074345,
-0.0368364080786705,
-0.031892213970422745,
-0.089454784989357,
-0.031196491792798042,
-0.019739601761102676,
0.01638079807162285,
0.01843675784766674,
0.0011201902525499463,
-0.01859862729907036,
0.018285078927874565,
-0.08277098089456558,
0.009562109597027302,
-0.03892478346824646,
-0.06959811598062515,
0.03379492834210396,
-0.009019732475280762,
-0.04053886607289314,
-0.01287792157381773,
-3.1684924927376736e-33,
-0.0519171841442585,
0.024590661749243736,
-0.04752172529697418,
0.001958083361387253,
-0.08128217607736588,
-0.0834011361002922,
0.11570349335670471,
0.018390005454421043,
0.05971340835094452,
0.04762407764792442,
0.03282729163765907,
0.005410967860370874,
-0.12837089598178864,
0.08993133157491684,
-0.006264807190746069,
-0.04913915693759918,
-0.053760427981615067,
-0.017692366614937782,
-0.003109204350039363,
-0.006901318207383156,
0.019694054499268532,
-0.07771388441324234,
0.03284817188978195,
-0.04738215357065201,
-0.030857015401124954,
0.01974380575120449,
0.05847007408738136,
-0.05514111742377281,
-0.001895503606647253,
0.014223809354007244,
-0.08324579894542694,
-0.05205103009939194,
0.08542013168334961,
0.06010498106479645,
-0.04442187026143074,
0.06894084811210632,
0.03227180615067482,
-0.024566244333982468,
0.028188439086079597,
-0.004929356276988983,
0.015593601390719414,
0.08164288848638535,
0.07830397039651871,
0.013910720124840736,
0.0036672865971922874,
0.07160092890262604,
0.05379435047507286,
0.13196684420108795,
-0.046509865671396255,
0.01114641409367323,
-0.047788478434085846,
-0.0630086213350296,
0.03680340573191643,
0.045211173593997955,
0.06917304545640945,
0.015901166945695877,
0.024980047717690468,
-0.06796203553676605,
0.02559376135468483,
-0.09057705849409103,
-0.0847325399518013,
-0.021201441064476967,
0.011583826504647732,
0.07187285274267197,
0.0604693666100502,
-0.093952976167202,
-0.06678890436887741,
-0.03379368036985397,
-0.015807252377271652,
-0.03293462097644806,
0.04166746512055397,
0.0261235311627388,
-0.058612506836652756,
-0.10887590795755386,
-0.012541417963802814,
-0.00014995652600191534,
-0.008820896036922932,
-0.04417041316628456,
-0.015572127886116505,
-0.0695226639509201,
-0.0017700925236567855,
0.021010931581258774,
0.02800842560827732,
0.07197748869657516,
-0.00187207933049649,
0.034791961312294006,
0.0298711396753788,
0.09423909336328506,
0.020638663321733475,
-0.0274258553981781,
0.11304237693548203,
0.06343796104192734,
-0.08817354589700699,
-0.03569521754980087,
0.12692905962467194,
8.556684927128489e-32,
-0.018180880695581436,
-0.044303275644779205,
-0.04676373302936554,
0.01987554132938385,
0.052972037345170975,
0.05971960723400116,
0.07026571035385132,
-0.009772381745278835,
0.006252251099795103,
-0.06589028239250183,
-0.03442370519042015,
0.033300552517175674,
-0.015554562211036682,
-0.005779316648840904,
0.021348590031266212,
0.03497910499572754,
-0.01637241430580616,
-0.03521641716361046,
-0.052669551223516464,
-0.028770117089152336,
-0.045115429908037186,
-0.018495142459869385,
-0.08467251062393188,
0.07937373220920563,
-0.05926332622766495,
0.04475647583603859,
-0.019985247403383255,
0.014942142181098461,
0.02184372954070568,
0.0063032107427716255,
-0.023948144167661667,
0.009609982371330261,
-0.019997822120785713,
-0.013148759491741657,
0.10310225933790207,
-0.017020437866449356,
-0.05081195384263992,
0.08583758026361465,
0.04092486947774887,
-0.06975600123405457,
-0.07203368097543716,
-0.005884452257305384,
0.01847386732697487,
0.0002750973217189312,
-0.0075589921325445175,
0.0012779913377016783,
0.02055838331580162,
-0.037010032683610916,
0.05164986476302147,
-0.014436603523790836,
-0.05391271039843559,
0.018775783479213715,
0.06612244993448257,
0.03435318544507027,
-0.012969952076673508,
-0.05983463302254677,
0.00774862477555871,
-0.046320755034685135,
0.07062411308288574,
0.05411108210682869,
-0.11782046407461166,
0.03415001183748245,
-0.013853915967047215,
-0.058616720139980316
] |
73Graphs and transformations
So the graph of y = g( x + 1) is
y
xy = g( x + 1)
= (x + 1)( x β 1)
1 β1β1O
β When you translate a function, any asymptotes are also translated.
Example 11
Given that h(x) = 1 __ x , sketch the curve with equation y = h(x) + 1 and state the equations of any
asymptotes and intersections with the axes
.
The graph of y = h( x) is
Oy
x1
xy =
So the graph of y = h( x) + 1 is
Oy
x1
The curve crosses the x -axis once.
y = h(x) +
1 = 1 __ x + 1
0 = 1 __ x + 1
β1 = 1 __ x
x = β1
So t
he curve intersects the x -axis at (β1, 0).
The horizontal asymptote is y = 1.
The vertical asymptote is x = 0.First sketch y = h(x ).
The curve is translated by vector ( 0 1 ) so the
asymptote is translated by the same vector.
Put y = 0 to find where the curve crosses the
x-axis.
Remember to write down the equation of the vertical asymptote as well. It is the y-axis so it has equation x = 0.This is a translation of the graph of y = g(x) by
vector ( β1 0 ) .
You could also write out the equation and sketch
the graph directly:
y = g(x
+ 1)
= (x + 1)(x + 1 β 2)
= (x + 1)(x β 1)
|
[
-0.0021682120859622955,
0.05173252895474434,
-0.011672512628138065,
-0.04665299504995346,
-0.10555767267942429,
-0.003565988503396511,
0.06433342397212982,
0.019084274768829346,
-0.027772502973675728,
-0.017244718968868256,
0.01330284122377634,
-0.010055229999125004,
-0.005291614681482315,
0.06108197569847107,
-0.06743825972080231,
0.0017463585827499628,
-0.048231128603219986,
0.009514939971268177,
-0.03822413459420204,
-0.09768257290124893,
0.06170196458697319,
-0.08655355125665665,
-0.04335813596844673,
-0.08221709728240967,
0.07510028779506683,
-0.06102140620350838,
-0.0027703428640961647,
-0.05574602633714676,
-0.01732390746474266,
-0.04150375723838806,
-0.03221522271633148,
0.032249823212623596,
0.028835058212280273,
-0.0021378041710704565,
0.06482736021280289,
0.03467728570103645,
0.0011140219867229462,
-0.01306330319494009,
0.024234501644968987,
-0.0014224420301616192,
-0.00723935104906559,
-0.007658916991204023,
-0.013342393562197685,
0.000028011152608087286,
0.0974937230348587,
-0.09382904320955276,
-0.05366738140583038,
0.01682977005839348,
0.023803628981113434,
0.0035978613886982203,
0.04687576740980148,
0.06640978157520294,
-0.0958278551697731,
0.036180127412080765,
-0.01618245802819729,
-0.037009432911872864,
-0.004538369830697775,
-0.026741694658994675,
0.015003379434347153,
0.019589127972722054,
-0.06056046485900879,
0.036958493292331696,
-0.07557167112827301,
0.04860718920826912,
-0.04839185252785683,
0.044808704406023026,
0.09031267464160919,
-0.031387850642204285,
0.010818567126989365,
0.10136663913726807,
-0.08382314443588257,
0.04278922826051712,
-0.021756241098046303,
-0.05233439430594444,
0.014889199286699295,
0.0036191169638186693,
0.04200882837176323,
0.09038814902305603,
-0.07740789651870728,
-0.047355882823467255,
-0.09106563031673431,
0.01890619657933712,
0.12258153408765793,
0.0503927543759346,
-0.08130937069654465,
-0.008647304959595203,
0.042107779532670975,
0.04505375027656555,
0.016190646216273308,
-0.07307539880275726,
-0.019600804895162582,
0.029199160635471344,
0.005664857104420662,
-0.09390965849161148,
0.010109895840287209,
-0.03000558167695999,
0.02530045434832573,
-0.020105483010411263,
0.0714380070567131,
0.04006287083029747,
-0.07260315865278244,
0.04450460150837898,
0.0867752805352211,
0.09940288215875626,
-0.023109404370188713,
-0.02393420971930027,
-0.02060464769601822,
0.01292372029274702,
0.0053944820538163185,
-0.06271733343601227,
-0.018136339262127876,
-0.045197494328022,
-0.021390872076153755,
0.03670240566134453,
0.06230848282575607,
-0.08690976351499557,
-0.021807007491588593,
-0.04455693066120148,
0.00457120593637228,
-0.005789881572127342,
0.05820637196302414,
0.016750691458582878,
0.09598741680383682,
0.03417475149035454,
-0.0559425912797451,
0.022434430196881294,
0.023460131138563156,
-0.00938620138913393,
0.025247322395443916,
-0.003174138953909278,
0.08405383676290512,
-0.0501784048974514,
0.05618949234485626,
0.0016224387800320983,
-0.018828479573130608,
0.00497974269092083,
-0.07437875866889954,
0.033746737986803055,
-0.02999952808022499,
0.058452095836400986,
0.038676753640174866,
0.11673755943775177,
-0.01724141649901867,
0.05022129788994789,
0.00806577317416668,
-0.04930293932557106,
-0.050618935376405716,
0.037057872861623764,
-0.03386348485946655,
0.023807961493730545,
-0.03948914259672165,
0.02496475726366043,
-0.04697981849312782,
0.08224714547395706,
0.0527629591524601,
-0.018616562709212303,
-0.05815016105771065,
-0.0632118359208107,
-0.0910167247056961,
-0.06108991801738739,
-0.06173298507928848,
-0.014427035115659237,
-0.06373301148414612,
0.044276028871536255,
0.061374641954898834,
-0.029902299866080284,
0.0087422551587224,
0.024203183129429817,
-0.016702307388186455,
0.009025313891470432,
0.052261222153902054,
-0.03597230464220047,
0.03209852799773216,
0.08171245455741882,
0.03981693461537361,
0.015651024878025055,
0.022488052025437355,
-0.02298055961728096,
-0.016828292980790138,
0.018621355295181274,
0.036105670034885406,
-0.06447885930538177,
0.02120203897356987,
-0.007561395410448313,
-0.04339764639735222,
-0.05577888712286949,
-0.034409407526254654,
0.0810466930270195,
0.000041307230276288465,
0.01625889353454113,
0.0010769630316644907,
-0.021995041519403458,
0.01904750056564808,
0.01862168498337269,
-0.08027294278144836,
-0.05368652939796448,
-0.018734557554125786,
0.03374257683753967,
-0.05480489879846573,
-0.0172701645642519,
-0.007791739422827959,
0.07671187818050385,
0.06575771421194077,
0.06214839592576027,
-0.04317350685596466,
-0.020278915762901306,
-0.03953328728675842,
-0.018896862864494324,
-0.09476567804813385,
-0.01150755025446415,
-0.0017345219384878874,
0.06288973987102509,
-0.027248498052358627,
0.05642322078347206,
0.011577397584915161,
-0.07058294117450714,
0.006289609707891941,
0.027902070432901382,
-0.05899728089570999,
-0.01684907265007496,
0.019639264792203903,
-0.07128705084323883,
0.05453922227025032,
9.558439559860311e-34,
-0.039397284388542175,
0.02310865744948387,
-0.061576686799526215,
-0.025092842057347298,
-0.004784437827765942,
-0.09515560418367386,
0.04553686082363129,
0.0547928661108017,
0.038325633853673935,
-0.005517000798135996,
0.10146168619394302,
0.01716446690261364,
-0.0752478837966919,
0.01010191161185503,
-0.026933126151561737,
-0.08294834196567535,
0.07469088584184647,
-0.02071930468082428,
-0.09246309846639633,
-0.0013869950780645013,
0.009045850485563278,
0.0021039238199591637,
0.052511997520923615,
-0.010900616645812988,
-0.11454501003026962,
-0.007425674702972174,
0.011687998659908772,
-0.04556279256939888,
0.0015980129828676581,
0.05513788014650345,
-0.006507613696157932,
-0.010795795358717442,
0.025494631379842758,
0.08959215134382248,
-0.028867747634649277,
0.06669086962938309,
-0.028916245326399803,
-0.04359808191657066,
0.009489893913269043,
0.03901039436459541,
0.033810805529356,
0.033460669219493866,
0.07671856880187988,
0.00455754017457366,
0.05542673543095589,
0.018015922978520393,
-0.03757566586136818,
0.09228929132223129,
-0.06508534401655197,
-0.028402462601661682,
-0.046496205031871796,
-0.08244451135396957,
0.027626218274235725,
-0.003155403770506382,
0.12230759114027023,
0.04013983532786369,
-0.021893328055739403,
-0.07423469424247742,
0.006535084452480078,
-0.11583815515041351,
-0.005314322654157877,
-0.01802893914282322,
-0.05175777152180672,
0.07964778691530228,
-0.08471279591321945,
-0.11625050753355026,
-0.052246447652578354,
-0.05988860875368118,
-0.04144681617617607,
-0.03313945233821869,
0.07339800894260406,
0.006804945413023233,
0.020239846780896187,
-0.033814687281847,
0.025420164689421654,
-0.018483391031622887,
-0.009836824610829353,
-0.06880161166191101,
0.020490285009145737,
-0.09862637519836426,
-0.008118730038404465,
0.039065562188625336,
-0.008229902014136314,
0.13848716020584106,
0.03597557544708252,
-0.03896082192659378,
0.06554257869720459,
0.08364304900169373,
0.07439687103033066,
0.0049858721904456615,
0.023345762863755226,
0.08211296051740646,
-0.0834224745631218,
-0.03118763118982315,
-0.01655529998242855,
6.361760732970835e-32,
-0.019280502572655678,
0.011455941945314407,
0.060110390186309814,
0.03495670109987259,
0.02003045380115509,
0.036893583834171295,
0.03564801812171936,
-0.09299785643815994,
0.012529649771749973,
-0.06827962398529053,
0.07307837903499603,
0.10668741166591644,
-0.08406843990087509,
0.061241019517183304,
0.021273983642458916,
-0.041145384311676025,
0.009658081457018852,
-0.0030530078802257776,
0.012371035292744637,
0.04587298631668091,
-0.06573987752199173,
0.024472253397107124,
-0.08407358080148697,
0.06694278866052628,
-0.0644640177488327,
-0.0030163791961967945,
-0.004881137516349554,
-0.07436849176883698,
0.00920439139008522,
-0.026372507214546204,
-0.023942310363054276,
0.04767671972513199,
0.060538556426763535,
-0.024080142378807068,
0.11514665186405182,
-0.0035579397808760405,
-0.12421728670597076,
0.04143934324383736,
0.03214579075574875,
-0.00729144923388958,
-0.08892402052879333,
0.011081707663834095,
0.02335486002266407,
0.027870425954461098,
-0.02812228910624981,
0.008207667618989944,
0.04470998793840408,
-0.03465142473578453,
0.06945033371448517,
-0.033241529017686844,
-0.06622710078954697,
-0.01018774975091219,
0.04162276163697243,
-0.12697923183441162,
-0.0023777044843882322,
0.030622271820902824,
0.043601762503385544,
0.005013014189898968,
-0.029224704951047897,
0.051920875906944275,
-0.04194622114300728,
0.1013139933347702,
-0.01984473317861557,
0.027702441439032555
] |
74
Chapter 4
1 Apply the f
ollowing transformations to the curves with equations y = f(x) where:
i f(x
) = x2 ii f(x) = x3 iii f(x) = 1 __ x
In each case state the coordina
tes of points where the curves cross the axes and in iii state the
equations of the asymptotes.
a f(x + 2) b f(x) + 2 c f(x
β 1)
d f(x)
β 1 e f(x
) β 3 f f(x
β 3)
2 a Sketch the curve
y = f(x) where f(x) = (x β 1)(x + 2).
b On separate dia
grams sketch the graphs of i y =
f(x + 2) ii y =
f(x) + 2.
c Find the equations of the curv
es y = f(x + 2) and y = f(x) + 2, in terms of x, and use these
equations to find the coordinates of the points where your graphs in part b cross the y-axis.
3 a Sketch the gra
ph of y = f(x) where f(x) = x2(1 β x).
b Sketch the curve with equa
tion y = f(x + 1).
c By finding the equation f(x
+ 1) in terms of x, find the coordinates of the point in part b
where the curve crosses the y-axis.
4 a Sketch the gra
ph of y = f(x) where f(x) = x(x β 2)2.
b Sketch the curves with equa
tions y = f(x) + 2 and y = f(x + 2).
c Find the coordinates of
the points where the graph of y = f(x + 2) crosses the axes.
5 a Sketch the gra
ph of y = f(x) where f(x) = x(x β 4).
b Sketch the curves with equa
tions y = f(x + 2) and y = f(x) + 4.
c Find the equations of the curv
es in part b in terms of x and hence find the coordinates of the
points where the curves cross the axes.
6 a Sketch the gra
ph of y = f(x) where f(x) = x2(x β 1)(x β 2).
b Sketch the curves with equa
tions y = f(x + 2) and y = f(x) β 1.
7 The point P(4,
β1) lies on the curve with equation y = f(x).
a State the coordina
tes that point P is transformed to on the curve with equation
y =
f(x β 2). (1 mark)
b State the coordina
tes that point P is transformed to on the curve with equation
y =
f(x) + 3. (1 mark)
8 The graph of
y = f(x) where f(x) = 1 __ x is translated so that the asymptotes are at x = 4 and
y = 0. Write down the equation for the transformed function in the form y = 1 _____ x + a (3 marks)
9 a Sketch the gra
ph of y = x3 β 5x2 + 6x, marking clearly the points of intersection with the axes.
b Hence sketch y
= (x β 2)3 β 5(x β 2)2 + 6(x β 2). E
E/P
PExercise 4E
|
[
-0.016780594363808632,
0.054968323558568954,
-0.08192957192659378,
-0.008509675040841103,
0.018274690955877304,
0.05182822793722153,
0.004271028097718954,
0.041193392127752304,
-0.12794603407382965,
-0.015121717937290668,
0.02162160724401474,
-0.07069893181324005,
0.03224297985434532,
0.035096921026706696,
-0.029587142169475555,
0.012667307630181313,
-0.13084736466407776,
0.015182237140834332,
-0.027190057560801506,
-0.03474820405244827,
0.013001156970858574,
-0.012953391298651695,
-0.06253288686275482,
-0.08579723536968231,
-0.016758011654019356,
-0.08086548745632172,
-0.06016836687922478,
-0.08496139198541641,
-0.031026283279061317,
-0.05376225709915161,
0.009939051233232021,
0.002767853904515505,
0.04833795130252838,
0.0005516057135537267,
0.07196111977100372,
-0.027190957218408585,
0.03248141333460808,
0.008910979144275188,
0.026711633428931236,
0.002081676386296749,
-0.0492670014500618,
0.05303505063056946,
-0.01854366809129715,
-0.027350693941116333,
0.1278456449508667,
-0.060951560735702515,
-0.015563622117042542,
0.024022923782467842,
-0.03632974252104759,
0.01679130643606186,
-0.01072030421346426,
-0.0022076070308685303,
-0.10634209960699081,
0.04667183384299278,
0.02848125994205475,
-0.01640038564801216,
-0.054152268916368484,
-0.003672997932881117,
-0.013015393167734146,
0.07824620604515076,
-0.017557434737682343,
0.06063318997621536,
-0.05217638984322548,
0.03147285431623459,
-0.07888460159301758,
0.07155263423919678,
0.10779083520174026,
-0.0516204908490181,
-0.012113773263990879,
0.1357155442237854,
-0.11152827739715576,
0.047448351979255676,
-0.05772444233298302,
-0.08880305290222168,
0.04622495546936989,
0.058441951870918274,
-0.004233370069414377,
0.06462559849023819,
-0.05243852734565735,
-0.018347224220633507,
-0.028943270444869995,
-0.07617945969104767,
0.06728526204824448,
0.037135351449251175,
-0.02772487886250019,
0.007237522397190332,
0.014369632117450237,
-0.07300017029047012,
0.020939111709594727,
-0.04041611775755882,
-0.075342558324337,
-0.029615184292197227,
-0.06469028443098068,
-0.005477324593812227,
-0.005136142484843731,
-0.05395715683698654,
-0.03625424951314926,
-0.04549267515540123,
0.10629929602146149,
0.10271049290895462,
-0.07801944762468338,
0.03686906397342682,
0.06438690423965454,
0.04640442505478859,
-0.05489800497889519,
-0.029574772343039513,
-0.00413513882085681,
-0.05863313004374504,
-0.05285244435071945,
-0.007693280465900898,
-0.019041603431105614,
-0.05947162210941315,
0.016140080988407135,
0.0194417554885149,
0.11159178614616394,
-0.0682673528790474,
0.0015951740788295865,
0.0460287407040596,
-0.022167233750224113,
-0.07491658627986908,
0.012012410908937454,
0.03669064864516258,
0.025312121957540512,
0.05439773201942444,
0.0023928030859678984,
-0.022348038852214813,
-0.029694633558392525,
0.00483397301286459,
-0.0004892182187177241,
-0.010036119259893894,
0.05328316614031792,
-0.10188817232847214,
-0.08846341073513031,
-0.06852011382579803,
0.019919227808713913,
0.026345396414399147,
-0.11723052710294724,
0.082240529358387,
-0.041343804448843,
0.0262894444167614,
0.05089300125837326,
0.05758120119571686,
0.0027236761525273323,
0.022420980036258698,
0.0965842753648758,
-0.029650112614035606,
-0.009373180568218231,
0.04154745489358902,
-0.00928241666406393,
0.016304848715662956,
0.01461738534271717,
-0.04494012892246246,
-0.012985012494027615,
0.10774217545986176,
0.08097287267446518,
0.013802903704345226,
-0.021673358976840973,
-0.02462473325431347,
-0.02523638866841793,
-0.028766347095370293,
-0.026695169508457184,
0.031605709344148636,
0.008833532221615314,
0.027377769351005554,
0.056816499680280685,
-0.057518452405929565,
0.015672920271754265,
0.10360883921384811,
-0.02939501777291298,
0.050230689346790314,
-0.009360096417367458,
-0.09178201109170914,
-0.005512642208486795,
0.11416361480951309,
0.05503474920988083,
0.07216053456068039,
0.1226930096745491,
-0.024140039458870888,
-0.017894001677632332,
-0.01257010456174612,
0.08410710841417313,
-0.048793964087963104,
0.0077636223286390305,
-0.023095592856407166,
-0.05816299095749855,
-0.06789299845695496,
0.02655082941055298,
0.030182702466845512,
0.02722799964249134,
-0.02235320396721363,
-0.028744664043188095,
-0.0935102179646492,
-0.0007419256144203246,
-0.02420196868479252,
-0.03171272203326225,
0.023339955136179924,
-0.007494446821510792,
0.010137954726815224,
-0.021442046388983727,
0.030963808298110962,
-0.03464096412062645,
0.0884261429309845,
0.04374028742313385,
0.04017239063978195,
-0.06420524418354034,
-0.029403941705822945,
-0.07691669464111328,
-0.10796181112527847,
-0.07434757053852081,
-0.050548236817121506,
-0.015116591937839985,
0.04693107306957245,
-0.0014238094445317984,
0.016065940260887146,
-0.03372989967465401,
-0.07047063112258911,
0.04444834217429161,
0.005200109910219908,
-0.07782569527626038,
0.03993226960301399,
0.021998243406414986,
-0.04198986291885376,
0.02658449485898018,
-9.530660607969465e-33,
-0.046927642077207565,
0.018694331869482994,
-0.08320052921772003,
0.03033898025751114,
-0.04907500743865967,
-0.08087703585624695,
0.017744962126016617,
0.02601597085595131,
0.00914521049708128,
0.022327981889247894,
0.09767990559339523,
-0.010651083663105965,
-0.05486714839935303,
-0.0379839725792408,
-0.03132782131433487,
-0.06738176941871643,
0.011334408074617386,
-0.031057020649313927,
-0.026154523715376854,
-0.006308292504400015,
0.03537442535161972,
0.03189442306756973,
0.021595362573862076,
0.0043707238510251045,
-0.12442608177661896,
0.016130920499563217,
-0.008385876193642616,
-0.018455056473612785,
-0.07941222190856934,
0.08398673683404922,
-0.05834439396858215,
-0.027120856568217278,
-0.02682812698185444,
-0.026684477925300598,
-0.024125682190060616,
0.0520387701690197,
-0.06773650646209717,
0.00040181903750635684,
0.018340226262807846,
0.030449002981185913,
0.016566935926675797,
0.09877367317676544,
0.0669671893119812,
-0.044256750494241714,
0.006795147899538279,
0.06633998453617096,
0.05994627624750137,
0.10529083013534546,
0.040887732058763504,
-0.004373841919004917,
-0.004141046665608883,
-0.09536509960889816,
-0.036224365234375,
-0.0026401057839393616,
-0.0014325006632134318,
0.05591517686843872,
0.019542960450053215,
-0.0932934507727623,
0.017919640988111496,
-0.10180789977312088,
-0.042998041957616806,
-0.02157926745712757,
-0.09834088385105133,
0.05265212431550026,
-0.021538808941841125,
-0.08099713921546936,
-0.08476296812295914,
-0.03095802664756775,
0.0324457623064518,
0.03607293590903282,
0.01710376888513565,
-0.05869432911276817,
-0.014254126697778702,
-0.0403694286942482,
-0.031077921390533447,
-0.008524246513843536,
0.0383891835808754,
-0.03144809231162071,
-0.003439090447500348,
-0.015848252922296524,
-0.024154234677553177,
0.010696292854845524,
-0.0064619798213243484,
0.0727529227733612,
-0.017841516062617302,
-0.004559845197945833,
0.0667230561375618,
0.07802629470825195,
0.06947696954011917,
0.013254315592348576,
-0.0024410556070506573,
-0.00333613995462656,
0.04850487783551216,
-0.031048940494656563,
0.0015506154159083962,
1.099842120490937e-31,
-0.09140627086162567,
0.00918411836028099,
0.015147312544286251,
0.0671669989824295,
0.09082625806331635,
0.04006058722734451,
0.03663447126746178,
-0.019652970135211945,
0.08247143775224686,
-0.10688818991184235,
0.08511463552713394,
0.10048303008079529,
-0.007427682168781757,
0.006537501700222492,
0.01390855386853218,
0.0004772824759129435,
-0.0008695857832208276,
0.028057800605893135,
-0.00658996170386672,
-0.02251817286014557,
-0.07711552828550339,
-0.01769375614821911,
-0.084018774330616,
-0.025063542649149895,
0.00616455078125,
-0.01684846542775631,
-0.027727488428354263,
-0.0844518318772316,
-0.038612380623817444,
0.06022557616233826,
0.012414983473718166,
-0.01821521855890751,
0.0765862986445427,
-0.0296090766787529,
0.014259962365031242,
-0.02073407731950283,
-0.017778538167476654,
0.030246173962950706,
-0.052200380712747574,
-0.012981534935534,
-0.00910111702978611,
0.029333150014281273,
-0.011518983170390129,
0.02161775529384613,
-0.04712434858083725,
0.020377948880195618,
-0.012820105068385601,
0.02638520672917366,
0.08034976571798325,
0.017266489565372467,
-0.004668102599680424,
0.05229039117693901,
0.017797335982322693,
-0.0729428380727768,
-0.03128048777580261,
0.022062575444579124,
0.02961939200758934,
-0.018414853140711784,
0.008100882172584534,
0.08533070981502533,
-0.11020521074533463,
0.0382402278482914,
-0.023419378325343132,
0.03262215480208397
] |
75Graphs and transformations
10 a Sketch the gra
ph of y = x2(x β 3)(x + 2), marking clearly the points of intersection with the axes.
b Hence sketch y
= (x + 2)2(x β 1)(x + 4).
11 a Sketch the gra
ph of y = x3 + 4x2 + 4x. (6 marks)
b The point with coordinates (
β1, 0) lies on the curve with
equation y = (x + a)3 + 4(x + a)2 + 4(x + a) where a is a
constant. Find the two possible values of a. (3 marks)
12 a Sketch the gra
ph of y = x(x + 1)(x + 3)2. (4 marks)
b Find the possible va
lues of b such that the point (2, 0) lies on the curve with equation
y = (x + b)(x + b + 1)(x + b + 3)2. (3 marks)P
E/P
E/P
1 Sketch the graph of y = (x β 3)3 + 2 and determine the coordinates of the point of inflection. β Section 12.9
2 The point Q (β5, β7) lies on the curve with equation y = f( x).
a Sta
te the coordinates that point Q is transformed to on the curve with equation y = f( x + 2) β 5.
b The c
oordinates of the point Q on a transformed curve are ( β3, β6). Write down the transformation in
the form y = f( x + a) β b.ChallengeLook at your sketch and
picture the curve sliding to the left or right.Problem-solving
4.6 Stretching graphs
Multiplying by a constant βoutsideβ the function stretches the graph vertically.
β The graph of y = af(x) is a stretch of the graph y = f(x) by a scale factor of a in the vertical
direction.
Oy
x β11y = f(x)1
2y = 2f(x)y = f(x)
Multiplying by a constant βinsideβ the function stretches the graph horizontally.
β The graph of y = f(ax) is a stretch of the graph y = f(x) by a scale factor of 1 __ a in the
horizontal direction.
Oy
x 2 64
y = f( x)1
3y = f(2x)
y = f(x)2f(x) is a stretch with scale factor 2 in the
y-direction. All y-coordinates are doubled.
1 _ 2 f(x) is a stretch with scale factor 1 _ 2 in the
y-direction.
All y-coordinates are halved.
y = f ( 1 _ 3 x) is a stretch with scale factor 3 in the
x-direction. All x-coordinates are tripled.y = f(2x) is a stretch with scale factor 1 _ 2 in the
x-direction.
All x-coordinates are halved.
|
[
-0.05483977869153023,
0.01898946799337864,
-0.0563930906355381,
-0.02154814451932907,
-0.040815871208906174,
0.03966899961233139,
-0.008281194604933262,
0.006534191779792309,
-0.06808431446552277,
0.01998276822268963,
0.020941298454999924,
-0.08860451728105545,
-0.020887719467282295,
-0.010503475554287434,
-0.05198042467236519,
-0.06421131640672684,
-0.008755536749958992,
-0.018291573971509933,
-0.029088323935866356,
-0.03856857493519783,
0.011972403153777122,
-0.07243043184280396,
-0.023628834635019302,
-0.09962072223424911,
0.10725220292806625,
-0.07089566439390182,
-0.03424632549285889,
-0.08148406445980072,
-0.04180237650871277,
-0.04729296267032623,
0.03401881456375122,
0.042947642505168915,
0.031175384297966957,
-0.05659105256199837,
0.07598918676376343,
0.010116771794855595,
0.021216822788119316,
0.018844690173864365,
0.07247116416692734,
0.006659215781837702,
0.016538379713892937,
0.029411116614937782,
-0.030958430841565132,
-0.06699348986148834,
0.09198135882616043,
-0.05623003467917442,
-0.039300959557294846,
-0.004503981210291386,
0.06946157664060593,
-0.07182686775922775,
0.042874667793512344,
0.007465626113116741,
-0.011209972202777863,
0.06928861141204834,
0.002653796225786209,
0.0303354449570179,
0.06727216392755508,
0.044656623154878616,
0.07360973954200745,
0.07538731396198273,
0.017970383167266846,
0.06256641447544098,
-0.028084328398108482,
0.042256392538547516,
-0.003992055542767048,
0.02988761104643345,
0.03889693319797516,
-0.0961628258228302,
-0.008653276599943638,
0.03815316781401634,
-0.07854847609996796,
0.01748751848936081,
-0.019518913701176643,
-0.12324454635381699,
0.006831106264144182,
-0.037937380373477936,
0.0013294549426063895,
0.006432714872062206,
-0.04547414183616638,
-0.0648905485868454,
-0.04293757304549217,
0.04314514994621277,
0.060458436608314514,
0.04658064246177673,
0.04410478100180626,
0.0050486731342971325,
0.03607694059610367,
-0.034138746559619904,
-0.06700826436281204,
0.004243042320013046,
-0.012120408937335014,
0.04349328204989433,
-0.05946745723485947,
-0.05125069618225098,
0.027652641758322716,
-0.1526317298412323,
0.0347481332719326,
-0.03387745842337608,
0.025738077238202095,
0.10571049898862839,
-0.046379219740629196,
0.09881319850683212,
0.013290205970406532,
-0.03654477745294571,
0.011080028489232063,
0.0018745441921055317,
0.04772872477769852,
-0.02438986487686634,
0.01380067691206932,
-0.01928708516061306,
-0.02042463608086109,
-0.03741725534200668,
0.02135670743882656,
0.07046960294246674,
0.04236387833952904,
-0.040150970220565796,
0.06634768843650818,
0.015554478392004967,
-0.01564430445432663,
-0.06296371668577194,
0.054895903915166855,
-0.0622110515832901,
-0.006007588002830744,
-0.025802655145525932,
-0.09107650071382523,
-0.022061733528971672,
-0.09800328314304352,
0.03259965404868126,
0.051368292421102524,
-0.026661448180675507,
0.08769591152667999,
-0.053445227444171906,
0.003771968884393573,
-0.042160455137491226,
-0.1189369484782219,
-0.05756261944770813,
-0.06380367279052734,
0.04376325383782387,
-0.059155385941267014,
-0.006060271989554167,
-0.0016116640763357282,
0.11082987487316132,
-0.001151396194472909,
0.07806618511676788,
0.01453503780066967,
0.0029304216150194407,
-0.07433062791824341,
0.030951984226703644,
0.02531852200627327,
0.004915285389870405,
-0.021403810009360313,
-0.004663569387048483,
-0.00046855496475473046,
0.06053929403424263,
0.04173116013407707,
-0.03439205139875412,
-0.04816557466983795,
-0.04299221187829971,
-0.022976232692599297,
0.009762129746377468,
-0.05568311735987663,
0.021177228540182114,
-0.04544806107878685,
0.045455556362867355,
0.044985853135585785,
-0.042309489101171494,
0.045522067695856094,
0.0923755094408989,
0.0000991787965176627,
0.08353284746408463,
-0.005019110161811113,
0.02708270028233528,
0.0012520182644948363,
0.13946130871772766,
0.016573714092373848,
-0.03458316996693611,
0.11692604422569275,
-0.003737900871783495,
-0.052540455013513565,
0.03344636783003807,
0.053999874740839005,
-0.031623370945453644,
-0.013658805750310421,
0.05523601919412613,
-0.030725596472620964,
-0.058943670243024826,
0.002144771860912442,
0.0656597837805748,
-0.0031223921105265617,
0.033920641988515854,
-0.04013213887810707,
-0.015216845087707043,
-0.030660366639494896,
0.114220529794693,
-0.1403340846300125,
-0.013933304697275162,
0.05953550338745117,
0.05880788341164589,
-0.0639999657869339,
0.002791397739201784,
0.020756473764777184,
0.037924401462078094,
0.04696124419569969,
-0.0036709345877170563,
-0.057137563824653625,
0.011584995314478874,
-0.045503418892621994,
0.012260973453521729,
-0.09260019659996033,
-0.022698624059557915,
0.07481937110424042,
0.05219675973057747,
-0.03038782626390457,
-0.010217631235718727,
0.022808894515037537,
-0.04946737736463547,
0.0427980050444603,
-0.07564753293991089,
-0.14039911329746246,
-0.024311672896146774,
0.012319764122366905,
-0.07029937207698822,
0.02418350800871849,
-2.9629242447177798e-33,
0.046747054904699326,
0.040077678859233856,
-0.04670938104391098,
-0.037831973284482956,
-0.019118493422865868,
-0.11024100333452225,
0.0989304631948471,
0.03343126177787781,
0.09477311372756958,
0.05538411810994148,
0.07653144747018814,
0.04026412591338158,
-0.015729637816548347,
0.03216945379972458,
0.012156701646745205,
-0.010543382726609707,
0.01296965777873993,
0.05108178034424782,
-0.02928212657570839,
-0.06649044156074524,
-0.003991518169641495,
0.007770549040287733,
0.01832597516477108,
0.01850380189716816,
-0.0134331239387393,
0.06395652890205383,
0.05137315019965172,
-0.12237374484539032,
-0.041325584053993225,
-0.023069720715284348,
-0.05969885364174843,
-0.056091342121362686,
0.0015859572449699044,
0.00965815782546997,
-0.04922039061784744,
0.0002104523591697216,
-0.015361416153609753,
-0.07769295573234558,
0.026572665199637413,
0.035873234272003174,
-0.002893269993364811,
0.04215312749147415,
0.0658150240778923,
0.023033538833260536,
-0.01458217017352581,
0.05522426217794418,
0.04458346217870712,
0.12344973534345627,
-0.036685194820165634,
0.022746378555893898,
-0.019330846145749092,
-0.0792132318019867,
-0.01465904526412487,
0.001282460754737258,
0.12957501411437988,
0.05920829996466637,
0.03330371528863907,
-0.014541742391884327,
-0.03126807138323784,
-0.0920269712805748,
-0.02851545624434948,
-0.035110484808683395,
-0.03862224519252777,
-0.016361616551876068,
-0.0632951483130455,
-0.07766810804605484,
-0.05232624337077141,
-0.06586422026157379,
0.011295280419290066,
-0.01828368380665779,
0.08494197577238083,
0.04431565850973129,
-0.014272896572947502,
-0.06696683168411255,
-0.0054712421260774136,
-0.10837550461292267,
-0.045267075300216675,
0.01120550837367773,
0.02056298404932022,
-0.04942669719457626,
-0.04901841655373573,
-0.07008369266986847,
-0.0006378481048159301,
0.05120024457573891,
0.006288607604801655,
0.019348423928022385,
0.020252397283911705,
0.07675027847290039,
0.12741287052631378,
-0.0038454767782241106,
0.06409952044487,
-0.004368661902844906,
-0.06392456591129303,
0.025800151750445366,
0.019474806264042854,
8.016328394217921e-32,
0.02714204043149948,
0.016876187175512314,
-0.01177150383591652,
0.00911396648734808,
-0.004964510910212994,
-0.027080930769443512,
0.07562822103500366,
-0.03695541247725487,
-0.0073236748576164246,
-0.061232153326272964,
0.05232306942343712,
0.014594298787415028,
-0.08178770542144775,
0.09458107501268387,
-0.06828419119119644,
-0.0015197376487776637,
-0.014898652210831642,
-0.005336912348866463,
-0.01505607832223177,
-0.04660209268331528,
-0.10168419033288956,
0.007295413874089718,
-0.035206958651542664,
0.06190827488899231,
-0.028568502515554428,
0.015376966446638107,
-0.05753168836236,
-0.10544361919164658,
-0.025508590042591095,
-0.013473778031766415,
-0.0029445372056216,
-0.031005486845970154,
0.007634075358510017,
-0.06956786662340164,
0.0518689788877964,
-0.03621921315789223,
-0.0779266208410263,
0.058432437479496,
0.011303769424557686,
0.008556139655411243,
0.0020091573242098093,
0.0009092472027987242,
-0.0386393740773201,
0.0064705428667366505,
-0.021496206521987915,
0.05252429470419884,
0.0341244600713253,
-0.02889876812696457,
0.06042703241109848,
-0.01602860912680626,
-0.05843166634440422,
0.05022360384464264,
0.05209873244166374,
-0.054419878870248795,
0.020067324861884117,
-0.022742098197340965,
0.0072182887233793736,
0.01871785894036293,
-0.007646489888429642,
0.05170981585979462,
0.010432968847453594,
0.09560077637434006,
-0.031087178736925125,
0.046029720455408096
] |
76
Chapter 4
Example 12
Given that f(x) = 9 β x2, sketch the curves with equations:
a y =
f(2x) b y =
2f(x)
a f(x) = 9 β x2
So f(x) = (3 β x)(3 + x )
The curve is y =
(3 β x )(3 + x )
0 = (3 β x )(3 + x )
So x = 3 or x = β 3
So the curve crosses the x -axis at (3, 0)
and (β3, 0).
When x = 0, y = 3 Γ 3 = 9
So the curve crosses the y -axis at (0, 9).
The curve y = f( x) is
3 β39
O xy
y = f(2 x) so the curve is
1.5 β1.59
O xy
b y = 2f(x) so the curve is
3 β3/one.ss018
x OyYou can factorise the expression.
Put y = 0 to find where the curve crosses the
x-axis.
Put x = 0 to find where the curve crosses the y-axis.
First sketch y = f(x).
y = f(ax) where a = 2 so it is a horizontal stretch with scale factor
1 _ 2 .
Check: The curve is
y = f(2x).
So y = (3 β 2x)(3 + 2x).
When y = 0, x = β1.5 or x = 1.5.So the curve crosses the x-axis at (β1.5, 0) and (1.5, 0).When x = 0, y = 9.So the curve crosses the y-axis at (0, 9).
y = af(x) where a = 2 so it is a vertical stretch with
scale factor 2.
Check: The curve is y = 2f(x).
So y = 2(3 β x)(3 + x).When y = 0, x = 3 or x = β3.So the curve crosses the x -axis at (β 3, 0) and (3, 0).
When x = 0, y = 2 Γ 9 = 18.So the curve crosses the y-axis at (0, 18).
|
[
0.011153441853821278,
0.11540050804615021,
-0.06174931675195694,
-0.02507524937391281,
0.025157272815704346,
0.04966996982693672,
-0.04062652215361595,
0.012362489476799965,
-0.08057485520839691,
0.022153962403535843,
0.016351742669939995,
-0.039625946432352066,
0.013299515470862389,
0.010953579097986221,
-0.036341723054647446,
-0.07060438394546509,
-0.12172470986843109,
0.008061633445322514,
-0.04969581216573715,
-0.0757938101887703,
-0.0046524349600076675,
-0.008378230966627598,
-0.09026814252138138,
-0.12061633169651031,
0.03463464975357056,
-0.10236313939094543,
-0.03278268501162529,
-0.02730332873761654,
-0.06748958677053452,
-0.0651356652379036,
0.06366581469774246,
0.011162914335727692,
-0.0011282303603366017,
0.032276540994644165,
0.06370251625776291,
0.03169172629714012,
0.03495018184185028,
-0.0010187451262027025,
0.03928912431001663,
-0.00786764919757843,
-0.04660814255475998,
-0.0058470843359827995,
0.009600224904716015,
-0.017307763919234276,
0.08161204308271408,
-0.03139515966176987,
0.005619919393211603,
0.011465199291706085,
0.07369054853916168,
-0.04714040085673332,
0.022431757301092148,
0.06644219160079956,
-0.0865817591547966,
0.03905100002884865,
0.018748614937067032,
0.007670286577194929,
0.03927714377641678,
0.030681144446134567,
0.01234729029238224,
0.08417822420597076,
0.012200049124658108,
0.013732141815125942,
-0.02420448698103428,
0.03404310345649719,
-0.015035992488265038,
0.07574643939733505,
-0.014588435180485249,
-0.07894370704889297,
-0.031746651977300644,
0.10802434384822845,
-0.12877513468265533,
0.021640969440340996,
-0.00016259281255770475,
-0.03198136389255524,
-0.019225096330046654,
-0.03700171783566475,
0.029492538422346115,
0.07026427239179611,
-0.06000413000583649,
-0.058476898819208145,
-0.06413523852825165,
-0.025073867291212082,
0.08284558355808258,
0.10238205641508102,
0.06146100535988808,
0.05733885243535042,
0.030596619471907616,
-0.07964476197957993,
0.013980933465063572,
0.012850688770413399,
-0.01985982060432434,
0.013000167906284332,
-0.07952403277158737,
0.027943165972828865,
0.01079781074076891,
-0.07659110426902771,
-0.04444127157330513,
-0.03439866751432419,
0.03222960606217384,
0.10786215960979462,
0.013062200509011745,
0.025740060955286026,
-0.007922869175672531,
0.08124913275241852,
0.02744365856051445,
-0.01314963586628437,
0.00496767507866025,
-0.024602266028523445,
-0.060371033847332,
-0.04706810414791107,
0.033071234822273254,
-0.0680476725101471,
0.003374592401087284,
-0.005546052940189838,
0.12751075625419617,
-0.07855839282274246,
0.017938781529664993,
-0.007422571070492268,
0.012905271723866463,
-0.07689987868070602,
0.03910205140709877,
-0.04200029373168945,
0.059740323573350906,
-0.046577610075473785,
-0.024380922317504883,
0.001467161695472896,
-0.014042823575437069,
-0.013941888697445393,
-0.009845558553934097,
-0.05286724865436554,
0.028460822999477386,
-0.09050628542900085,
-0.04821009561419487,
-0.05439239740371704,
-0.002095377305522561,
-0.004870667587965727,
-0.15595823526382446,
0.15196266770362854,
0.04455491900444031,
-0.02527480386197567,
0.030238162726163864,
0.03789878264069557,
0.06333161890506744,
-0.009428056888282299,
0.04134991019964218,
-0.04410732910037041,
-0.044525131583213806,
-0.008021255023777485,
-0.015828611329197884,
0.036401502788066864,
0.03538565710186958,
-0.036426398903131485,
0.020101133733987808,
0.10706507414579391,
0.01409014593809843,
0.005621445830911398,
0.002455496694892645,
-0.05797422304749489,
-0.014944014139473438,
-0.06475212424993515,
-0.016622714698314667,
0.05889594554901123,
-0.0472877211868763,
0.06107693910598755,
0.018789106979966164,
0.024754386395215988,
0.0515831895172596,
0.026698337867856026,
0.004954793490469456,
-0.031094007194042206,
0.01886998862028122,
0.028162416070699692,
-0.008980275131762028,
0.0762878954410553,
0.04032421484589577,
0.04963575676083565,
0.0697268396615982,
0.01667151041328907,
-0.0022502276115119457,
0.0037410135846585035,
0.12171746790409088,
-0.07235094159841537,
0.00959502812474966,
-0.05421896651387215,
-0.01060173287987709,
-0.009984211064875126,
-0.014111470431089401,
0.01915138028562069,
0.03221631050109863,
0.016401084139943123,
-0.031741999089717865,
-0.010599440895020962,
-0.052016645669937134,
0.08306040614843369,
-0.0909143015742302,
-0.015878062695264816,
0.06527207791805267,
0.08612436801195145,
-0.06329020857810974,
0.05855702981352806,
-0.009216382168233395,
-0.03010551445186138,
0.09813430905342102,
0.04338856413960457,
-0.09371865540742874,
0.019157877191901207,
0.011967101134359837,
0.006400546990334988,
0.04440366476774216,
0.045544009655714035,
-0.09191424399614334,
0.026739120483398438,
-0.028736261650919914,
0.021555639803409576,
-0.020335284993052483,
-0.05052267014980316,
0.0724744126200676,
-0.023934468626976013,
-0.12694884836673737,
-0.004712569527328014,
-0.06835176795721054,
0.0030658342875540257,
0.02567971684038639,
1.9951957653384468e-33,
-0.034892674535512924,
0.07754329591989517,
-0.08584776520729065,
0.01740238443017006,
-0.03407565504312515,
-0.014164835214614868,
0.10991374403238297,
0.033751871436834335,
0.04652886092662811,
0.06513393670320511,
-0.01029881089925766,
0.028208307921886444,
-0.08033058792352676,
-0.0038570850156247616,
-0.019002873450517654,
-0.05339209362864494,
-0.03780142217874527,
-0.08609714359045029,
0.0063882265239953995,
-0.06440377235412598,
0.03110305219888687,
0.02303994633257389,
-0.018607379868626595,
-0.019150545820593834,
-0.01657191850244999,
0.007128527853637934,
0.06236329674720764,
-0.07971344143152237,
-0.03444649651646614,
-0.034902431070804596,
-0.06336880475282669,
-0.03582775965332985,
0.05290338769555092,
0.03021211363375187,
-0.021613914519548416,
0.03972082585096359,
-0.049335695803165436,
0.011232882738113403,
0.03821681812405586,
-0.0061797951348125935,
0.09149830788373947,
0.08036363124847412,
0.051915787160396576,
-0.04369689151644707,
-0.0143679678440094,
0.11150114983320236,
-0.011727232486009598,
0.08003739267587662,
0.005020199343562126,
0.028236428275704384,
-0.030693883076310158,
-0.1257370561361313,
0.056436847895383835,
0.009873216040432453,
0.05074281990528107,
0.0030322729144245386,
0.011554224416613579,
-0.13275308907032013,
0.013119888491928577,
-0.021514704450964928,
-0.051076650619506836,
0.004029375966638327,
-0.03190857172012329,
0.09478685259819031,
0.005816091317683458,
-0.030011897906661034,
-0.0713334009051323,
-0.0632801204919815,
0.07922237366437912,
0.016274496912956238,
0.01251231785863638,
0.12783953547477722,
-0.002052419353276491,
-0.1584606021642685,
-0.04552159458398819,
-0.03579295054078102,
-0.04983818531036377,
0.01834379881620407,
0.06974869966506958,
-0.10660205781459808,
0.001226358232088387,
0.006889128126204014,
-0.053097087889909744,
0.036705125123262405,
-0.02666645310819149,
0.009007811546325684,
-0.016520729288458824,
0.06700891256332397,
0.05189478024840355,
0.017147595062851906,
0.08614541590213776,
-0.018845349550247192,
-0.06467972695827484,
-0.08639736473560333,
0.07037560641765594,
8.336128685325229e-32,
-0.06473524868488312,
-0.006704185623675585,
-0.01611975207924843,
0.01768028363585472,
-0.0009110012906603515,
0.016013989225029945,
0.07861505448818207,
-0.022478749975562096,
0.018572891131043434,
-0.005088589619845152,
0.027990106493234634,
0.03269296884536743,
-0.029047120362520218,
0.039674002677202225,
0.032910022884607315,
0.024866925552487373,
0.015189453028142452,
-0.05607152357697487,
0.006115178111940622,
-0.015048363246023655,
-0.05338360741734505,
-0.034467730671167374,
-0.06262193620204926,
0.024775763973593712,
0.052078649401664734,
0.021708739921450615,
0.012078365311026573,
-0.018473505973815918,
0.0050497837364673615,
-0.08071893453598022,
-0.0056737507693469524,
-0.009307353757321835,
0.05010196566581726,
-0.030391275882720947,
0.05676295608282089,
-0.042647913098335266,
0.003299757605418563,
0.015697989612817764,
-0.045761458575725555,
-0.016268471255898476,
0.008323829621076584,
-0.04689393192529678,
0.019691508263349533,
-0.021323880180716515,
-0.021928604692220688,
-0.05165707692503929,
-0.01781930774450302,
0.026344506070017815,
0.01807487942278385,
0.00647767074406147,
-0.015236573293805122,
0.01860622689127922,
0.007247673813253641,
-0.0234372615814209,
0.018542194738984108,
-0.004378174897283316,
0.02254306711256504,
-0.007171242032200098,
-0.023544201627373695,
-0.0473276749253273,
-0.11852888762950897,
0.08921805024147034,
-0.025594783946871758,
-0.01632414385676384
] |
77Graphs and transformations
Example 13
a Sketch the curve with equation y = x(x β 2)(x + 1).
b On the same axes, sk
etch the curves y = 2x(2x β 2)(2x + 1) and y = βx(x β 2)(x + 1).
a
Oy
x 2y = x(x β 2)( x + 1)
β1
b
y = x(x β 2)( x + 1)y = 2 x(2x β 2)(2 x + 1)y = βx(x β 2)( x + 1)
Oy
x 2 β1y = βx(x β 2)(x + 1) is a stretch with scale factor
β1 in the y-direction. Notice that this stretch has the effect of reflecting the curve in the x-axis.
y = 2x(2x β 2)(2x + 1) is a stretch with scale factor
1 _ 2 in the x-dir ection.
You need to work out the relationship between
each new function and the original function. If
x(x β 2)(x + 1) = f( x) then
2x(2x β 2)(2 x + 1) = f(2 x), and
βx(x β 2)(x + 1) = β f(x).Problem-solving
β The graph of y = βf(x) is a reflection of the graph of y = f(x) in the x-axis.
β The graph of y = f(β x) is a reflection of the graph of y = f(x) in the y-axis.
Example 14
On the same axes sketch the graphs of y = f(x),
y = f(βx) and y = βf(x) where f(x) = x(x + 2).
f(x) = x(x + 2)
y
x O 2 β2
y = βf(x)y = f( x) y = f(βx)y = f(βx) is y = (βx)(βx + 2) which is y = x2 β 2x
or y = x(x β 2) and this is a reflection of the
original curve in the y-axis.
Alternatively multiply each x-coordinate by β1
and leave the y-coordinates unchanged.
This is the same as a stretch parallel to the x-axis
scale factor β1.
y = βf(x) is y = βx(x + 2) and this is a reflection of the original curve in the x-axis.
Alternatively multiply each y-coordinate by β1
and leave the x-coordinates unchanged.
This is the same as a stretch parallel to the y-axis
scale factor β1. Explore stretches of the graph
of
y = x (x β 2)( x + 1) using GeoGebra.Online
|
[
-0.08607317507266998,
-0.013573684729635715,
-0.030756376683712006,
-0.07673896849155426,
-0.03592035174369812,
0.05368461459875107,
-0.055394891649484634,
-0.008264830335974693,
-0.07694945484399796,
-0.04549087956547737,
-0.005500972270965576,
-0.025399621576070786,
0.01709596998989582,
-0.05032398924231529,
-0.08668231964111328,
-0.02525498904287815,
-0.02656141109764576,
-0.01846071146428585,
-0.02549111284315586,
-0.042502060532569885,
0.0249753687530756,
-0.1108175739645958,
-0.035765063017606735,
-0.08129974454641342,
0.11118119210004807,
-0.08943918347358704,
0.001714429585263133,
-0.04398924857378006,
-0.01331798080354929,
-0.06744489073753357,
-0.0011559419799596071,
0.04281020537018776,
0.015311921946704388,
0.011925587430596352,
0.08527541905641556,
-0.0066968235187232494,
0.027166131883859634,
0.023102618753910065,
0.03667300194501877,
-0.005095070693641901,
-0.004239234142005444,
0.017067840322852135,
0.027911311015486717,
-0.02763601951301098,
0.0609247200191021,
-0.026606708765029907,
-0.05020726099610329,
-0.0284752044826746,
0.06004232540726662,
-0.027754174545407295,
0.06340168416500092,
0.0474073700606823,
-0.014188675209879875,
0.007879804819822311,
-0.06213697791099548,
0.022734494879841805,
-0.006668010260909796,
0.04395470768213272,
0.029787475243210793,
0.006909498479217291,
-0.004583005793392658,
0.0334942564368248,
-0.0291744377464056,
0.06519141048192978,
-0.033641260117292404,
0.056022390723228455,
0.044885680079460144,
-0.042962636798620224,
0.019209641963243484,
0.03714120388031006,
-0.08046755939722061,
0.05304897204041481,
-0.06106964126229286,
-0.07009661942720413,
0.01140409242361784,
-0.061022497713565826,
0.002009423915296793,
0.05458056554198265,
-0.006947381887584925,
-0.10940540581941605,
-0.0910959541797638,
-0.05627748370170593,
0.09429635852575302,
0.043522804975509644,
-0.011053070425987244,
-0.00915861688554287,
0.06389674544334412,
0.018065961077809334,
-0.035033877938985825,
0.031995050609111786,
0.001112287980504334,
0.030561937019228935,
-0.01405235193669796,
-0.09580117464065552,
0.042206913232803345,
-0.12940141558647156,
0.035168904811143875,
0.007437800522893667,
0.07645902037620544,
0.06011698767542839,
0.01125869620591402,
0.04738190397620201,
0.029313305392861366,
0.09294486790895462,
0.012783440761268139,
-0.05755852162837982,
-0.056168220937252045,
-0.01155303418636322,
-0.007357548922300339,
-0.011791515164077282,
-0.05327539145946503,
-0.06910084933042526,
-0.01014809962362051,
0.06861218065023422,
0.015878649428486824,
-0.09680277109146118,
0.036968182772397995,
0.0299705658107996,
0.024698356166481972,
-0.03442384675145149,
0.055705856531858444,
-0.002557370811700821,
0.04097963124513626,
-0.00264963461086154,
-0.063993439078331,
-0.01648111455142498,
-0.01555212028324604,
0.00723837548866868,
-0.003912916872650385,
-0.04033292084932327,
0.08412332832813263,
-0.005599703174084425,
0.0424395389854908,
-0.02085793949663639,
-0.03391271457076073,
-0.04566902667284012,
-0.0400819331407547,
0.07603954523801804,
-0.03085668385028839,
0.010349801741540432,
0.0123054888099432,
0.09565328061580658,
0.06524579226970673,
0.06222720444202423,
-0.009518477134406567,
-0.02893664874136448,
-0.10306688398122787,
0.017670391127467155,
0.028993478044867516,
0.040596261620521545,
-0.05128234997391701,
-0.05339420959353447,
-0.0025256199296563864,
0.08923278748989105,
0.07204937189817429,
-0.06718175113201141,
-0.04675962030887604,
-0.054907966405153275,
-0.061875488609075546,
-0.03067697398364544,
-0.04477567970752716,
0.05147923156619072,
-0.06260059773921967,
0.03450248762965202,
0.051291875541210175,
-0.04109867662191391,
-0.001690313103608787,
0.08118417859077454,
0.025976093485951424,
0.03184042498469353,
0.011018434539437294,
0.042698170989751816,
-0.0055611697025597095,
0.1280505657196045,
0.02498319000005722,
-0.019468287006020546,
0.09140992909669876,
-0.028019970282912254,
-0.057911600917577744,
-0.0006382517167367041,
0.07349355518817902,
-0.014160492457449436,
0.02323845587670803,
0.030443822965025902,
-0.07178143411874771,
-0.05629661679267883,
-0.04998233541846275,
0.0570659302175045,
0.01879267767071724,
0.04540617763996124,
-0.0567176379263401,
-0.0431453213095665,
0.0018044939497485757,
0.09361707419157028,
-0.10432988405227661,
-0.031205862760543823,
0.03517819195985794,
0.015004732646048069,
-0.09883257001638412,
0.021018011495471,
0.05871133878827095,
0.08965450525283813,
0.09130171686410904,
0.031226105988025665,
-0.019532812759280205,
-0.033797603100538254,
-0.05161614716053009,
-0.02996811829507351,
-0.08130760490894318,
0.025225037708878517,
0.024818599224090576,
0.05813445523381233,
-0.03413233906030655,
0.030561594292521477,
-0.02015675976872444,
-0.10246118903160095,
0.020367087796330452,
-0.04060108959674835,
-0.035862404853105545,
-0.0038586114533245564,
-0.025749841704964638,
-0.09557703137397766,
0.037922605872154236,
-2.7141099268731186e-33,
-0.025499368086457253,
0.017166098579764366,
-0.0662568062543869,
0.016708333045244217,
-0.03500239923596382,
-0.14061222970485687,
0.08348773419857025,
0.010526226833462715,
0.0673358365893364,
0.01372387632727623,
0.058513518422842026,
-0.002596555044874549,
-0.0785636156797409,
-0.03366130590438843,
-0.014874226413667202,
-0.024583876132965088,
0.03383354842662811,
-0.01071570347994566,
-0.044048111885786057,
-0.03209250420331955,
0.030385656282305717,
0.00803808681666851,
0.021202439442276955,
0.003522430080920458,
-0.055180326104164124,
0.06172117218375206,
0.06160981208086014,
-0.08477379381656647,
0.018013840541243553,
0.029811998829245567,
-0.07650242000818253,
-0.07690747082233429,
0.053263600915670395,
0.053276702761650085,
-0.08670774847269058,
0.04476764425635338,
-0.011413304135203362,
-0.01770944707095623,
0.0436614453792572,
0.057750869542360306,
0.031234199181199074,
0.04193181172013283,
0.027768075466156006,
0.02869483269751072,
0.03461555764079094,
0.03842264786362648,
0.006490804720669985,
0.15276020765304565,
-0.014643044210970402,
0.0028433268889784813,
-0.020450320094823837,
-0.06067950651049614,
0.01860644854605198,
-0.0009884119499474764,
0.11563795059919357,
0.026161234825849533,
-0.010722311213612556,
-0.05361369252204895,
0.006880756467580795,
-0.09731761366128922,
0.001941403141245246,
-0.011733971536159515,
-0.03234146535396576,
0.04850127547979355,
-0.046632930636405945,
-0.08734745532274246,
-0.04580296203494072,
-0.06830542534589767,
-0.0298266913741827,
-0.04379692301154137,
0.10573678463697433,
0.04114019498229027,
-0.019675159826874733,
-0.07214974611997604,
-0.0021411629859358072,
-0.07934053987264633,
-0.008624903857707977,
0.007180246990174055,
-0.014351587742567062,
-0.01928703673183918,
-0.05459558963775635,
0.01621459424495697,
0.0228870902210474,
0.046829648315906525,
0.0176851823925972,
0.009113923646509647,
0.09723136574029922,
0.07743671536445618,
0.07283276319503784,
0.02211572602391243,
0.05605144798755646,
-0.035529062151908875,
-0.013601789250969887,
0.008649487048387527,
0.03946773707866669,
7.734223268205606e-32,
0.019975515082478523,
0.011906719766557217,
0.022977661341428757,
-0.0000044781363612855785,
0.010909420438110828,
0.032822396606206894,
0.02511577308177948,
-0.01707376539707184,
0.00880503561347723,
-0.07115913182497025,
0.07399621605873108,
0.13186520338058472,
-0.09070879220962524,
0.08941122144460678,
0.030546382069587708,
0.033982519060373306,
-0.02739250846207142,
-0.0234705600887537,
-0.02055671438574791,
-0.033817049115896225,
-0.12605522572994232,
-0.03766431286931038,
-0.04798157885670662,
0.06774543225765228,
-0.10207488387823105,
-0.011560287326574326,
-0.07438582926988602,
-0.0883399173617363,
-0.021509071812033653,
0.015217664651572704,
-0.023458046838641167,
0.02112882025539875,
0.04936103895306587,
-0.053160134702920914,
0.08216951787471771,
-0.013098057359457016,
-0.10507994890213013,
0.07338283210992813,
0.026950562372803688,
0.0026437335181981325,
-0.022617366164922714,
-0.02384340390563011,
-0.012681723572313786,
0.06964771449565887,
-0.01189162116497755,
0.025276266038417816,
0.05103554204106331,
-0.007700510788708925,
0.023570619523525238,
-0.07943928241729736,
-0.04244600236415863,
0.009867043234407902,
0.05719301849603653,
-0.03485502675175667,
-0.006811192259192467,
-0.0503375381231308,
-0.04306546971201897,
0.03598993644118309,
0.052169911563396454,
0.006003060843795538,
-0.05400766432285309,
0.05371911823749542,
-0.05543886125087738,
0.03190308064222336
] |
78
Chapter 4
1 Apply the f
ollowing transformations to the curves with equations y = f(x) where:
i f(x
) = x2 ii f(x) = x3 iii f(x) = 1 __ x
In each case show both f(x
) and the transformation on the same diagram.
a f(2x) b f(βx
) c f( 1 _ 2 x) d f(4x) e f( 1 _ 4 x)
f 2f(x
) g βf(x
) h 4f(x
) i 1 _ 2 f(x) j 1 _ 4 f(x)
2 a Sketch the curv
e with equation y = f(x) where f(x) = x2 β 4.
b Sketch the gra
phs of y = f(4x), 1 _ 3 y = f(x), y = f(βx) and y = βf(x).
3 a Sketch the curv
e with equation y = f(x) where f(x) = (x β 2)(x + 2)x.
b Sketch the gra
phs of y = f( 1 _ 2 x), y = f(2x) and y = βf(x).
4 a Sketch the curv
e with equation y = x2(x β 3).
b On the same axes, sk
etch the curves with equations:
i y =
(2x)2(2x β 3) ii y = βx2(x β 3)
5 a Sketch the curv
e y = x2 + 3x β 4.
b On the same axes, sk
etch the graph of 5y = x2 + 3x β 4.
6 a Sketch the gr
aph of y = x2(x β 2)2.
b On the same axes, sk
etch the graph of 3y = βx2(x β 2)2.
7 The point P(2,
β3) lies on the curve with equation y = f(x).
a State the coordina
tes that point P is transformed to on the curve with equation
y = f(2x). (1 mark)
b State the coordina
tes that point P is transformed to on the curve with equation
y = 4f(x). (1 mark)
8 The point Q(
β2, 8) lies on the curve with equation y = f(x).
State the coor dina
tes that point Q is transformed to on the curve with equation
y = f( 1 _ 2 x). (1 mark)
9 a Sketch the gr
aph of y = (x β 2)(x β 3)2. (4 marks)
b The graph of
y = (ax β 2)(ax β 3)2 passes through the point (1, 0).
Find two possible values for a . (3 marks) For part b, rearrange
the s
econd equation into
the form y = 3f( x).Hint
P
Let f( x) = x2(x β 3) and try to
write each of the equations
in part b in terms of f( x).Problem-solving
E
E
E/PExercise 4F
1 The point R(4, β 6) lies on the curve with equation y = f(x). State the coordinates
that point R is transformed to on the curve with equation y = 1 _ 3 f(2x).
2 The p
oint S(β4, 7) is transformed to a point S9(β8, 1.75). Write down the
transformation in the form y = af(bx).Challenge
|
[
-0.008617733605206013,
0.025739924982190132,
-0.10825558006763458,
-0.049027103930711746,
0.015980763360857964,
0.08586245775222778,
0.014097458682954311,
0.004709181841462851,
-0.08724911510944366,
-0.029178299009799957,
0.017304832115769386,
-0.06409555673599243,
0.011816606856882572,
0.0232030488550663,
-0.027653643861413002,
-0.0038238416891545057,
-0.13632717728614807,
0.00967564806342125,
-0.04411696642637253,
-0.05983884632587433,
0.0532958097755909,
-0.06238788366317749,
-0.08259832113981247,
-0.11338648200035095,
-0.00605771504342556,
-0.065727099776268,
-0.06009244546294212,
-0.03766130656003952,
-0.031202808022499084,
-0.08028767257928848,
0.025446079671382904,
0.019673295319080353,
0.0321209616959095,
0.050074853003025055,
0.0619288794696331,
0.001921256771311164,
-0.009696044027805328,
0.0045525566674768925,
0.05210462585091591,
-0.020780475810170174,
-0.0670974850654602,
0.04816848039627075,
-0.0068291970528662205,
-0.03289323300123215,
0.0726814940571785,
-0.029898211359977722,
0.022754130885004997,
0.019455919042229652,
-0.024798167869448662,
0.02501826360821724,
0.0028971524443477392,
-0.03695223852992058,
-0.09620721638202667,
-0.018418217077851295,
0.061123643070459366,
0.01902008056640625,
0.013700383715331554,
0.07645200937986374,
0.0005395451444201171,
0.05329686030745506,
-0.02211267128586769,
0.05833938345313072,
-0.005568814929574728,
0.04715876281261444,
-0.06756109744310379,
0.07371950149536133,
0.05858966335654259,
-0.046950362622737885,
-0.04359722137451172,
0.11227791011333466,
-0.1097753569483757,
0.05125013366341591,
-0.10080159455537796,
-0.08014184236526489,
0.03305434435606003,
0.07162847369909286,
0.014310481958091259,
0.10335967689752579,
-0.061449985951185226,
-0.04313897714018822,
-0.024100787937641144,
-0.09666410833597183,
0.08621950447559357,
-0.014834389090538025,
-0.005463933572173119,
0.04769285023212433,
-0.04438355565071106,
-0.09021066874265671,
0.013966181315481663,
-0.007965855300426483,
-0.08220865577459335,
-0.014934172853827477,
-0.04631851613521576,
-0.05399180203676224,
-0.010540693067014217,
-0.06878983974456787,
-0.05791645124554634,
-0.011306816712021828,
0.10512463003396988,
0.06671018153429031,
-0.08425161242485046,
-0.005123198498040438,
0.1426295042037964,
0.05508648231625557,
-0.05893164873123169,
-0.016300152987241745,
-0.06342071294784546,
-0.030608804896473885,
-0.06634287536144257,
0.013917088508605957,
0.03885738179087639,
-0.04851479083299637,
-0.0012875152751803398,
0.0031367784831672907,
0.08564697206020355,
-0.08165494352579117,
-0.02269309014081955,
0.07821618765592575,
-0.024023206904530525,
-0.06575952470302582,
-0.028333932161331177,
0.023029401898384094,
0.0015022136503830552,
0.05783764645457268,
-0.009807122871279716,
-0.06986548751592636,
-0.027708491310477257,
-0.005834984593093395,
-0.024384619668126106,
-0.015028699301183224,
0.02470579743385315,
-0.04924895241856575,
-0.07440274208784103,
-0.06893383711576462,
0.015817854553461075,
-0.0013845048379153013,
-0.11894428730010986,
0.07744095474481583,
-0.03610479459166527,
0.013744491152465343,
0.0026864896062761545,
0.07986602932214737,
0.03632959723472595,
0.024453017860651016,
0.0768430307507515,
-0.025778472423553467,
-0.027893511578440666,
0.01754835620522499,
0.027937844395637512,
0.05838119238615036,
-0.04935217276215553,
-0.07157053053379059,
-0.011954644694924355,
0.07548199594020844,
0.015043006278574467,
-0.015162416733801365,
-0.037476640194654465,
-0.014555827714502811,
0.006671702954918146,
-0.036212190985679626,
-0.016429590061306953,
0.03782404586672783,
0.03587932512164116,
0.0031038776505738497,
0.045406728982925415,
-0.05713865906000137,
0.0012677961494773626,
0.12692669034004211,
-0.018908794969320297,
-0.0003782019193749875,
0.03453437238931656,
-0.11257076263427734,
0.017673010006546974,
0.12220703810453415,
0.06395626813173294,
0.0917418822646141,
0.09924324601888657,
-0.07275322824716568,
-0.040843214839696884,
-0.026264790445566177,
0.0753268152475357,
-0.07026020437479019,
0.013076373375952244,
-0.038857851177453995,
-0.0432746559381485,
-0.0019137754570692778,
0.02399687096476555,
0.06750630587339401,
-0.005562968552112579,
-0.024101898074150085,
-0.03229339420795441,
-0.021956324577331543,
-0.00736126396805048,
-0.00774924922734499,
-0.008808314800262451,
0.013666572980582714,
-0.011869235895574093,
0.03991309925913811,
-0.03502647578716278,
0.030834045261144638,
-0.015970513224601746,
0.07764822244644165,
0.10351830720901489,
0.027396127581596375,
-0.058019351214170456,
-0.02371428906917572,
-0.04395940154790878,
-0.11707381904125214,
-0.10756617784500122,
0.023677516728639603,
-0.0338263101875782,
0.025149613618850708,
-0.0004692135553341359,
0.0382765531539917,
-0.0037682149559259415,
-0.02890358306467533,
0.06072413921356201,
0.05270915850996971,
-0.04990531504154205,
0.04825817048549652,
-0.013156807981431484,
-0.016218174248933792,
0.05839582160115242,
-7.12891773367623e-33,
-0.062126077711582184,
0.03863785043358803,
-0.059262268245220184,
0.06847267597913742,
-0.04237954691052437,
-0.023961102589964867,
0.014659358188509941,
0.009322162717580795,
0.03908830136060715,
0.014171412214636803,
0.07653800398111343,
-0.034565236419439316,
-0.06664474308490753,
-0.061319269239902496,
-0.05229513719677925,
-0.08422260731458664,
-0.012331304140388966,
0.003536459058523178,
-0.04251660034060478,
-0.012063325382769108,
0.03446974232792854,
0.03350627422332764,
0.024115730077028275,
0.030283113941550255,
-0.11465433984994888,
0.01097732875496149,
0.0312572717666626,
0.02761639654636383,
-0.03738643229007721,
0.056780919432640076,
-0.05578348785638809,
-0.018166033551096916,
-0.004709594417363405,
-0.041240084916353226,
-0.01056615263223648,
0.04284954443573952,
-0.03737761452794075,
0.040403302758932114,
-0.0034312857314944267,
0.020998720079660416,
0.010465200990438461,
0.09715965390205383,
0.021344151347875595,
-0.025931308045983315,
-0.018822211772203445,
0.05603118613362312,
0.07837490737438202,
0.061327945441007614,
0.010970951057970524,
-0.024335769936442375,
-0.009978294372558594,
-0.09860335290431976,
-0.01785503327846527,
-0.06495934724807739,
-0.03793967142701149,
0.06921495497226715,
0.040586329996585846,
-0.10055892914533615,
-0.007754484191536903,
-0.10707157850265503,
-0.03615030273795128,
0.014227756299078465,
-0.08115614205598831,
0.07814904302358627,
-0.015590636059641838,
-0.12340983748435974,
-0.08699560165405273,
-0.04161418229341507,
0.02076072245836258,
0.05632975697517395,
0.030489612370729446,
-0.008454927243292332,
0.006122257094830275,
-0.008791097439825535,
0.026922347024083138,
-0.03554490953683853,
0.05593343824148178,
-0.010937875136733055,
-0.02869349718093872,
0.025755181908607483,
-0.01851409487426281,
-0.000390549743315205,
0.0056047760881483555,
0.03075183555483818,
0.010599828325212002,
-0.01301606371998787,
0.06713934242725372,
0.05740921199321747,
0.037087857723236084,
0.013429708778858185,
-0.01028421800583601,
0.020717915147542953,
0.06316138803958893,
0.002010082360357046,
-0.043059997260570526,
1.0324129433413285e-31,
-0.09281370043754578,
0.006119589786976576,
0.017547806724905968,
0.01577613316476345,
0.08503307402133942,
0.007824067026376724,
0.08513209223747253,
0.004401755053550005,
0.028477005660533905,
-0.058711741119623184,
0.06903856247663498,
0.15453992784023285,
-0.0017487788572907448,
0.02055744268000126,
0.04826854169368744,
0.02506866678595543,
-0.009846706874668598,
-0.002846313873305917,
-0.030106283724308014,
-0.013355785980820656,
-0.061609260737895966,
-0.051453907042741776,
-0.052666373550891876,
0.02857683040201664,
-0.026043085381388664,
-0.04012419655919075,
-0.034280743449926376,
-0.0785529688000679,
-0.04054800420999527,
0.03422151133418083,
0.00018351881590206176,
-0.02311960980296135,
0.0527021586894989,
-0.05265878885984421,
0.012521459721028805,
-0.052855025976896286,
-0.04471663385629654,
0.033756010234355927,
-0.04489381983876228,
0.02054172195494175,
-0.003996338229626417,
0.04032100364565849,
0.05337739735841751,
0.05121690407395363,
-0.03777186572551727,
0.03719601407647133,
0.03279383108019829,
0.010564999654889107,
0.004271449986845255,
0.03717762604355812,
0.02090277709066868,
-0.006185542326420546,
0.035863205790519714,
-0.061983536928892136,
-0.023159874603152275,
-0.009447176940739155,
0.06230900436639786,
-0.019589034840464592,
0.04107973724603653,
0.06729989498853683,
-0.09461276978254318,
0.007313606794923544,
0.003601975506171584,
-0.007718183100223541
] |
79Graphs and transformations
4.7 Transforming functions
You can apply transformations to unfamiliar functions by considering how specific points and features
are transformed.
Example 15
The following diagram shows a sketch of the curve f(x) which passes through the origin. The points A(1, 4) and B(3, 1) also lie on the curve.
Sketch the following:
a
y =
f(x + 1) b y =
f(x β 1) c y =
f(x) β 4
d 2y
= f(x) e y β
1 = f(x)
In each case you should show the positions of the images
of the points O, A and B.y
x Oy = f(x)
B (3, 1)A (1, 4)
a f(x + 1)
O(2, 1)/four.ss01
β1 xy
y = f( x + 1)
b f(x
β 1)
O(/four.ss01, 1)
1(2, /four.ss01)
xy
y = f( x β 1)
c f(x) β
4
y
x O
β/four.ss011
(3, β3)y = f( x) β 4Translate f(x) 1 unit in the direction of the
negative x-axis.
Translate f(x) 1 unit in the direction of the positive x-axis.
Translate f(x) 4 units in the direction of the negative y-axis.
|
[
-0.016790170222520828,
0.02158244140446186,
-0.05481753498315811,
-0.07096415013074875,
-0.04640055075287819,
0.04527558386325836,
-0.01431749016046524,
0.0012071575038135052,
-0.011572487652301788,
-0.03747863322496414,
0.0028491325210779905,
-0.015574891120195389,
-0.027858257293701172,
0.012413457967340946,
-0.03636348247528076,
-0.04458470642566681,
-0.02888108417391777,
0.04044263809919357,
-0.0011150246718898416,
-0.06355589628219604,
0.03638070449233055,
-0.0781404972076416,
-0.05823112279176712,
-0.1697525531053543,
0.07277736812829971,
-0.0807027816772461,
0.022219084203243256,
-0.013011621311306953,
-0.07374771684408188,
-0.10234642028808594,
0.0018587177619338036,
0.009021628648042679,
0.011511058546602726,
0.023452432826161385,
0.06438083946704865,
-0.028754794970154762,
-0.0242579597979784,
0.05013692006468773,
0.06253049522638321,
0.007562537211924791,
0.026349512860178947,
0.06038500741124153,
0.05272921174764633,
-0.012915479019284248,
0.06481341272592545,
-0.06780384480953217,
0.0029248534701764584,
0.010143091902136803,
0.055544037371873856,
-0.05430107191205025,
0.04330515116453171,
0.034039005637168884,
-0.054811082780361176,
-0.000503552844747901,
0.08566374331712723,
0.057626836001873016,
0.07587374746799469,
0.05600912496447563,
0.02680305205285549,
0.046043071895837784,
0.0009188978583551943,
0.061960771679878235,
-0.03464015573263168,
-0.0032052581664174795,
-0.02946837805211544,
0.1132500022649765,
0.054398611187934875,
-0.053359322249889374,
-0.0021136384457349777,
0.07821536064147949,
-0.1318519562482834,
0.02447544038295746,
-0.05619365721940994,
-0.04500735178589821,
0.012719007208943367,
-0.016942953690886497,
0.04957036301493645,
0.11628692597150803,
-0.06114238500595093,
-0.0641583651304245,
-0.05026204138994217,
-0.008228588849306107,
0.08952342718839645,
0.03260830044746399,
0.01137311290949583,
0.05310256406664848,
0.00019159021030645818,
-0.040115635842084885,
0.027463948354125023,
-0.004382250364869833,
-0.08674705773591995,
0.04049777612090111,
-0.033089570701122284,
-0.09057152271270752,
-0.005266835913062096,
-0.10440319031476974,
0.0013067509280517697,
0.0033464087173342705,
0.10287639498710632,
0.04964178055524826,
-0.06622392684221268,
0.07999785244464874,
0.045514341443777084,
0.06693138182163239,
-0.04275404289364815,
-0.013182376511394978,
-0.0543438084423542,
-0.014726629480719566,
0.0003006989136338234,
-0.05610504001379013,
0.001893355161882937,
-0.03487930819392204,
-0.08624601364135742,
0.05525054410099983,
0.053935740143060684,
-0.1049630343914032,
-0.02355375699698925,
-0.02181079052388668,
0.0180618017911911,
-0.00826498307287693,
0.037548549473285675,
0.027966827154159546,
0.010480217635631561,
0.01417940016835928,
-0.10122371464967728,
0.032421715557575226,
-0.06955830752849579,
-0.02802979201078415,
0.031159156933426857,
0.029969926923513412,
0.08916405588388443,
-0.025344252586364746,
0.004528042860329151,
-0.03448835015296936,
-0.04899809509515762,
-0.02951877750456333,
-0.06557770818471909,
0.04002809897065163,
-0.019159313291311264,
0.0813073068857193,
0.01004522480070591,
0.1062568724155426,
0.0859544426202774,
0.02283991314470768,
0.019115786999464035,
-0.04193073883652687,
-0.07362735271453857,
0.000478549423860386,
0.048881784081459045,
-0.006805435288697481,
-0.001982530113309622,
-0.008687661029398441,
-0.016433555632829666,
0.0652398020029068,
0.012068871408700943,
-0.007457928266376257,
-0.06625142693519592,
-0.05658029392361641,
-0.02680124342441559,
-0.03242524340748787,
-0.03656252846121788,
0.0012589701218530536,
-0.01143031008541584,
0.05203995853662491,
0.08512042462825775,
-0.020325902849435806,
0.05750978738069534,
0.09391530603170395,
0.032914094626903534,
0.05023912340402603,
-0.029741184785962105,
-0.08090721815824509,
0.01151781901717186,
0.09239113330841064,
0.001400456763803959,
0.03611122816801071,
0.06252233684062958,
0.003521040314808488,
-0.049289360642433167,
-0.017618665471673012,
0.047421324998140335,
-0.07397498935461044,
0.02617526426911354,
0.018189093098044395,
-0.031989604234695435,
-0.0865025445818901,
-0.019293591380119324,
0.02844937890768051,
0.00014540096162818372,
-0.001559077063575387,
0.0081110168248415,
-0.008773870766162872,
-0.013929840177297592,
-0.02151845395565033,
-0.11322464793920517,
-0.0030751600861549377,
0.04893133416771889,
0.029035748913884163,
-0.02178768627345562,
0.017485763877630234,
-0.009797273203730583,
0.02258593589067459,
0.14433704316616058,
0.075999416410923,
-0.07139983028173447,
-0.032867614179849625,
-0.019749963656067848,
-0.023767102509737015,
-0.03428587689995766,
0.00007042381912469864,
0.013617205433547497,
0.057158321142196655,
0.00797678530216217,
-0.01698736660182476,
-0.009678499773144722,
-0.120700903236866,
0.07135030627250671,
-0.00722018675878644,
-0.06269245594739914,
-0.005691241007298231,
-0.013986362144351006,
-0.040945738554000854,
0.04525407776236534,
-3.0314546470157575e-33,
-0.00562323909252882,
0.05213294178247452,
-0.04192344471812248,
0.008626498281955719,
-0.05852847173810005,
-0.08180275559425354,
0.06361240893602371,
0.014752866700291634,
0.07455068826675415,
0.05216439440846443,
-0.04088209569454193,
0.022440047934651375,
-0.10953176766633987,
-0.03058217279613018,
-0.03660353273153305,
-0.05635622888803482,
0.00038873188896104693,
-0.03031846135854721,
-0.11108226329088211,
-0.02347254753112793,
0.033900804817676544,
0.03382435068488121,
-0.01716412790119648,
-0.025683054700493813,
-0.105317622423172,
0.013140188530087471,
0.07915638387203217,
-0.02600100263953209,
-0.005887956358492374,
0.014647538773715496,
-0.09348927438259125,
-0.03105468861758709,
0.07013628631830215,
0.04537321254611015,
-0.04766551032662392,
0.055059097707271576,
-0.004644750151783228,
-0.01613609306514263,
0.05144964158535004,
0.011971307918429375,
0.012059307657182217,
0.06314285844564438,
0.04937157779932022,
0.007223423570394516,
0.03267499431967735,
0.05301375687122345,
0.03604205325245857,
0.10194902122020721,
-0.01922355778515339,
0.004000394139438868,
-0.06763955950737,
-0.06843994557857513,
0.012079199776053429,
-0.01155718881636858,
0.08218028396368027,
0.03134167566895485,
0.03127265349030495,
-0.08675143122673035,
0.014142770320177078,
-0.07778625190258026,
-0.04688600078225136,
-0.0348745658993721,
-0.05022188276052475,
0.028740454465150833,
-0.07621566951274872,
-0.1379527747631073,
-0.07521384954452515,
-0.03826315701007843,
-0.04007291421294212,
-0.054703082889318466,
0.11537062376737595,
0.04845711961388588,
0.00731884827837348,
-0.05857344716787338,
0.024595772847533226,
-0.05805308744311333,
0.017628392204642296,
-0.022154856473207474,
-0.048837609589099884,
0.03985943645238876,
0.009848187677562237,
-0.02502640336751938,
-0.015477685257792473,
0.08119223266839981,
0.017024947330355644,
0.00040615221951156855,
0.04535304754972458,
0.08957953006029129,
0.05649333447217941,
0.037011612206697464,
-0.01965835504233837,
0.016511734575033188,
-0.02519250474870205,
-0.011125367134809494,
-0.0045163813047111034,
7.773444224967967e-32,
0.001406054012477398,
0.05386639013886452,
0.016007015481591225,
0.04786806181073189,
0.0006607057293877006,
-0.03768913820385933,
0.08130458742380142,
-0.02381783537566662,
-0.019731910899281502,
-0.031218791380524635,
0.02038727141916752,
0.13081540167331696,
-0.02322910726070404,
0.07645468413829803,
0.038714680820703506,
-0.009243895299732685,
0.039329513907432556,
-0.08256795257329941,
-0.03831589221954346,
-0.02786354534327984,
-0.07466550916433334,
0.015707938000559807,
-0.04553358629345894,
0.05050861835479736,
-0.022828228771686554,
-0.0039887456223368645,
-0.04322083666920662,
-0.03360065072774887,
0.011294220574200153,
-0.05166105180978775,
-0.04952665790915489,
0.0038735889829695225,
0.07929807901382446,
-0.06988505274057388,
0.06544878333806992,
-0.022700104862451553,
-0.08283122628927231,
0.019649449735879898,
-0.031582895666360855,
0.01393604651093483,
0.017799120396375656,
0.0020264936611056328,
-0.0251668281853199,
0.019725767895579338,
-0.0797397717833519,
0.019275344908237457,
0.06252086162567139,
-0.005979551468044519,
-0.01531313732266426,
-0.07299023121595383,
-0.03836441412568092,
0.010426326654851437,
0.04449935629963875,
-0.12401161342859268,
-0.0038439808413386345,
0.03135642036795616,
0.04852332919836044,
-0.02072589471936226,
0.019241156056523323,
0.07720605283975601,
-0.06851031631231308,
0.055934708565473557,
0.013904096558690071,
0.006277456413954496
] |
80
Chapter 4
d 2y = f(x) so y = 1 __ 2 f(x)
(3, )(1, 2)
1
2y = f(x)1
2y
x O
e y β 1 = f( x) so y = f( x) + 1
1y
x O(3, 2)(1, 5)
y = f( x) + 1Rearrange in the form y = β¦
Stretch f(x) by scale factor 1 _ 2 in the y-dir ection.
Rearrange in the form y = β¦
Translate f(x) 1 unit in the direction of the
positive y-axis.
1 The following diagram shows a sketch of the curve y
x O(4, 4)
C
B DA2
1 6
with equation y = f(x). The points A(0, 2), B(1, 0),
C(4, 4) and D(6, 0) lie on the curve.
Sketch the following graphs and give the coordinates
of the points, A, B, C and D after each transformation:
a f(x
+ 1) b f(x
) β 4 c f(x
+ 4)
d f(2x
) e 3f(x
) f f( 1 _ 2 x)
g 1 _ 2 f(x) h f(β x)
2 The curve y
= f(x) passes through the origin and y
x Ox = 1y = 2
has horizontal asymptote y = 2 and vertical
asymptote x = 1, as shown in the diagram.
Sketch the following graphs. Give the equations of
any asymptotes and give the coordinates of intersections with the axes after each transformation.
a
f(x
) + 2 b f(x
+ 1) c 2f(x
)
d f(x
) β 2 e f(2x
) f f( 1 _ 2 x)
g 1 _ 2 f(x) h βf(x )Exercise 4G
|
[
-0.047331757843494415,
0.03014230914413929,
-0.01388527825474739,
0.003610537853091955,
-0.005678805988281965,
0.09377336502075195,
0.06767749786376953,
-0.0213429294526577,
0.005796169862151146,
0.005806676112115383,
0.008952684700489044,
-0.013110651634633541,
0.015286065638065338,
0.05770239606499672,
-0.016853921115398407,
-0.03748529776930809,
-0.09528159350156784,
0.04476425051689148,
-0.018394576385617256,
-0.07852808386087418,
0.0701611116528511,
0.011257040314376354,
-0.14395838975906372,
-0.05105414614081383,
0.03614501655101776,
0.013265758752822876,
-0.04189402610063553,
-0.013699527829885483,
-0.06018689274787903,
-0.10345479100942612,
0.01848086342215538,
0.04422146454453468,
-0.05535726621747017,
0.06994406133890152,
0.038832228630781174,
0.03510737046599388,
0.06356263905763626,
-0.00822303257882595,
0.029768269509077072,
-0.05016719922423363,
-0.048026226460933685,
0.020060425624251366,
-0.02278929203748703,
0.0047823358327150345,
0.06479861587285995,
-0.002373764058575034,
-0.0485035665333271,
0.06972486525774002,
0.06831822544336319,
-0.05279644951224327,
0.003094827989116311,
0.041437193751335144,
-0.09585981070995331,
-0.030345642939209938,
0.08210489898920059,
-0.06257247924804688,
0.07275021821260452,
-0.021823683753609657,
-0.02873295359313488,
0.07211983948945999,
-0.0653780847787857,
0.007675219792872667,
0.00849162507802248,
0.012591551057994366,
-0.016447078436613083,
0.1051839143037796,
-0.041658706963062286,
-0.0400884710252285,
-0.08078163862228394,
0.10499628633260727,
-0.12821845710277557,
-0.019960317760705948,
-0.02478562667965889,
-0.04648980870842934,
0.1081710159778595,
0.037020985037088394,
-0.02456422708928585,
0.030339641496539116,
0.048723552376031876,
0.030472731217741966,
-0.030765678733587265,
-0.039814598858356476,
0.0405627004802227,
0.0056022643111646175,
-0.049396589398384094,
0.005746550392359495,
-0.035263583064079285,
-0.03892632946372032,
0.007345321588218212,
-0.014908910728991032,
0.01997729204595089,
0.008496221154928207,
-0.028667356818914413,
0.0015044895699247718,
-0.0022085970267653465,
-0.08635500818490982,
0.006874291691929102,
-0.04807579889893532,
0.010821118950843811,
0.12593337893486023,
-0.052577659487724304,
0.045408908277750015,
0.03819376975297928,
0.018573999404907227,
-0.05376831814646721,
-0.06501149386167526,
0.04721478745341301,
-0.059397198259830475,
0.0474168136715889,
-0.015779437497258186,
0.007745671551674604,
-0.03393402323126793,
-0.05984881892800331,
0.05788815766572952,
0.08207916468381882,
-0.017902912572026253,
0.027730878442525864,
-0.06988868862390518,
0.0065012420527637005,
-0.08075415343046188,
0.040775783360004425,
-0.042978812009096146,
-0.03337045758962631,
0.003047540318220854,
-0.03510279580950737,
-0.08659016340970993,
0.03786979615688324,
-0.013565798290073872,
0.03596179932355881,
0.009468144737184048,
-0.0036847905721515417,
-0.04932081326842308,
0.03868778795003891,
-0.008000236004590988,
-0.03136522322893143,
0.042249977588653564,
-0.13594520092010498,
0.026582932099699974,
-0.012039913795888424,
0.027502914890646935,
0.005738539155572653,
0.06820473819971085,
0.07926058024168015,
-0.07709329575300217,
0.04495217278599739,
0.029590371996164322,
0.05643778294324875,
-0.0003848455671686679,
0.04449526593089104,
0.0807260274887085,
0.02925153076648712,
0.010594256222248077,
-0.03587808832526207,
-0.014864839613437653,
0.025190390646457672,
-0.035409022122621536,
-0.03385182470083237,
-0.05615746229887009,
-0.06410801410675049,
-0.02809533290565014,
-0.08653419464826584,
-0.035335227847099304,
-0.04457509145140648,
0.04347442090511322,
0.03349067643284798,
0.0411544069647789,
0.0009927208302542567,
0.11788876354694366,
-0.044621605426073074,
0.008380185812711716,
0.07104043662548065,
-0.04814975708723068,
0.015497086569666862,
0.0696435421705246,
-0.013852089643478394,
0.02981622703373432,
0.1364569514989853,
0.047914404422044754,
0.006015002727508545,
0.0277862586081028,
0.09578177332878113,
-0.07895391434431076,
-0.009493660181760788,
0.014770381152629852,
-0.039275139570236206,
-0.13970068097114563,
-0.09328978508710861,
0.06758761405944824,
-0.031587518751621246,
0.032456107437610626,
0.0464707612991333,
0.027532992884516716,
-0.041054606437683105,
0.020824402570724487,
-0.043026186525821686,
-0.009165175259113312,
0.012344478629529476,
0.03924284875392914,
-0.053843263536691666,
0.04192521423101425,
0.004030311480164528,
0.07440657168626785,
0.08145011961460114,
0.011722750030457973,
0.05208302289247513,
-0.026938991621136665,
-0.07893974334001541,
-0.07340428233146667,
-0.08499867469072342,
-0.038663674145936966,
0.014715521596372128,
0.03740783408284187,
-0.004132071975618601,
-0.0071772984229028225,
0.008051971904933453,
-0.10366431623697281,
0.03508700057864189,
0.038166116923093796,
-0.07011765241622925,
0.0008569892379455268,
0.08835189044475555,
-0.04708871617913246,
0.012806938961148262,
-2.5086577589203684e-33,
-0.056566022336483,
0.06145340949296951,
-0.05416443571448326,
-0.023149291053414345,
-0.0336633026599884,
-0.026835652068257332,
0.028572160750627518,
0.016540369018912315,
0.07566598057746887,
0.07522042840719223,
-0.009175926446914673,
0.009392454288899899,
-0.019109057262539864,
-0.06566176563501358,
-0.07331831008195877,
-0.049393218010663986,
-0.05139024183154106,
0.024384750053286552,
-0.05444461107254028,
0.061349984258413315,
0.049936357885599136,
-0.0351191908121109,
-0.004315918777137995,
0.01821722462773323,
-0.09650449454784393,
0.011140203103423119,
0.02074216865003109,
-0.027530191466212273,
0.007016260642558336,
0.05951336398720741,
-0.09687779098749161,
-0.03876190632581711,
0.04114184528589249,
0.03304986655712128,
-0.008462442085146904,
-0.00042503775330260396,
-0.013027042150497437,
-0.019502421841025352,
0.049150675535202026,
-0.005610258784145117,
0.06361062079668045,
0.04660555347800255,
-0.004971650894731283,
-0.06100115180015564,
0.08145734667778015,
0.02969708852469921,
0.03767433390021324,
0.017542356625199318,
0.015450102277100086,
-0.0635564774274826,
-0.06403060257434845,
-0.06672350317239761,
-0.03034607507288456,
0.09277071803808212,
0.0035300222225487232,
0.012220467440783978,
0.04920708388090134,
-0.02092580683529377,
-0.006906058173626661,
-0.027798334136605263,
-0.05989985540509224,
-0.010674132965505123,
-0.04582119360566139,
0.03101471997797489,
-0.005016045644879341,
-0.109431691467762,
-0.08364438265562057,
0.01270067784935236,
-0.06576190888881683,
0.04270917549729347,
0.04378755763173103,
0.010129790753126144,
0.002572237281128764,
0.007908191531896591,
-0.043606437742710114,
-0.01728106662631035,
-0.03799829259514809,
-0.01851639710366726,
-0.04070718213915825,
-0.0025175544433295727,
0.0006710559246130288,
0.05044206604361534,
0.006178900133818388,
0.033585648983716965,
-0.004940299782902002,
0.012740397825837135,
0.12652023136615753,
0.05973353981971741,
0.06172095984220505,
-0.02381269820034504,
-0.04188881069421768,
0.04019855335354805,
-0.006086569745093584,
-0.06560014933347702,
0.0204518660902977,
9.079657074084746e-32,
-0.07551703602075577,
-0.08323153853416443,
-0.02040092647075653,
0.07256053388118744,
0.11958853900432587,
0.00876945722848177,
0.05375847592949867,
0.03514338657259941,
-0.02064051665365696,
-0.08336284011602402,
0.009298713877797127,
0.0643441453576088,
-0.020775513723492622,
0.10994115471839905,
-0.03801726922392845,
-0.02090262807905674,
0.006597532425075769,
0.019264444708824158,
-0.09477895498275757,
-0.032624103128910065,
-0.019520673900842667,
0.001994726248085499,
-0.12333478033542633,
-0.040653351694345474,
0.0031121561769396067,
-0.00091176712885499,
-0.033037248998880386,
-0.043018825352191925,
-0.07098358869552612,
0.019409269094467163,
-0.020083079114556313,
-0.005808803718537092,
-0.06920602172613144,
-0.10654664039611816,
0.013285702094435692,
0.05415002629160881,
-0.0659731850028038,
0.09591550379991531,
-0.06783479452133179,
0.012081382796168327,
-0.08104915171861649,
0.015022710897028446,
-0.0023662999738007784,
0.013711064122617245,
-0.023648587986826897,
0.001673602731898427,
0.018075549975037575,
-0.05864777788519859,
0.048761650919914246,
-0.02686203271150589,
-0.017484314739704132,
-0.03839440643787384,
0.0707966536283493,
-0.005411764606833458,
-0.04546277970075607,
0.020560581237077713,
0.04354194924235344,
0.017872804775834084,
-0.01271376945078373,
0.07856833189725876,
-0.056129105389118195,
0.08506154268980026,
0.07358260452747345,
-0.026662487536668777
] |
81Graphs and transformations
3 The curve with equation
y = f(x) passes through the y
x OB
CD
A (β4, β6)β2 4
β3
points A(β4, β6), B(β2, 0), C(0, β3) and D(4, 0)
as shown in the diagram.
Sketch the following and give the coordinates of
the points A, B, C and D after each transformation.
a f(x
β 2) b f(x
) + 6 c f(2x
)
d f(x
+ 4) e f(x
) + 3 f 3f(x
)
g 1 _ 3 f(x) h f( 1 _ 4 x) i βf(x )
j f(β
x)
4 A sketch of the curv
e y = f(x) is shown in the y
x Ox = β21
diagram. The curve has a vertical asymptote
with equation x = β2 and a horizontal asymptote with equation y = 0. The curve crosses the y-axis at (0, 1).
a
Sketch, on separa
te diagrams, the
graphs of:
i 2f(x
) ii f(2x
) iii f(x
β 2)
iv f(x
) β 1 v f(β
x) vi βf(x
)
In each case state the equations of any
asymptotes and, if possible, points where the curve cuts the axes.
b
Suggest a possible equation f
or f(x).
5 The point P(2, 1) lies on the gr
aph with equation y = f(x).
a On the graph of
y = f(ax), the point P is mapped to the point Q(4, 1).
Determine the value of a. (1 mark)
b Write down the coor
dinates of the point to which P maps under each transformation
i f(x
β 4) ii 3f(x
) iii 1 _ 2 f(x) β 4 (3 marks)
6 The diagram sho
ws a sketch of a curve with equation y = f(x).
The points A(β1, 0), B(0, 2), C(1, 2) and D(2, 0) lie on the curve.
Sketch the following graphs and give the coordinates of the points
A, B, C and D after each transformation:
a y +
2 = f(x) b 1 _ 2 y = f(x)
c y β
3 = f(x) d 3y
= f(x)
e 2y
β 1 = f(x)E/P
OABC
Dy
xP
Rearrange each equation
into the form y = β¦Problem-solving
|
[
-0.011856946162879467,
0.02744477614760399,
-0.042537104338407516,
-0.04090229421854019,
-0.01198847871273756,
0.06760215014219284,
-0.014975075609982014,
-0.05927043408155441,
-0.0823395624756813,
-0.008436844684183598,
-0.006486008875072002,
-0.04300575330853462,
-0.0010384637862443924,
-0.010492788627743721,
-0.08475387841463089,
-0.02103600464761257,
-0.06090943515300751,
-0.009680572897195816,
-0.06456082314252853,
-0.038375530391931534,
0.04894215241074562,
-0.05532343313097954,
-0.000533567275851965,
-0.17328369617462158,
0.10332532227039337,
-0.06039079278707504,
0.03463560715317726,
0.007359621115028858,
-0.08451784402132034,
-0.06922915577888489,
0.003326472593471408,
-0.015147583559155464,
0.02913452498614788,
-0.005877169780433178,
0.1013421043753624,
-0.02897169068455696,
-0.01713920757174492,
0.043118916451931,
0.07728346437215805,
-0.049431413412094116,
0.0035365389194339514,
0.01484060287475586,
0.0154538219794631,
-0.005963907577097416,
0.0912506952881813,
-0.021682053804397583,
-0.03424633666872978,
0.02249431051313877,
0.07779262214899063,
-0.030104773119091988,
0.035668518394231796,
0.04204905033111572,
-0.035489536821842194,
0.016349809244275093,
0.06325293332338333,
0.010719955898821354,
0.07035607844591141,
0.049871884286403656,
0.054135825484991074,
0.050236523151397705,
0.012857426889240742,
0.08834468573331833,
-0.03662300854921341,
0.027953198179602623,
-0.02965417131781578,
0.14895179867744446,
0.07433199882507324,
-0.07888273149728775,
-0.01838316023349762,
0.06117436662316322,
-0.09723007678985596,
-0.005204659420996904,
-0.09251173585653305,
-0.041954487562179565,
0.046685751527547836,
-0.01731245592236519,
0.009721554815769196,
0.07418043911457062,
0.010736086405813694,
-0.05395878106355667,
-0.019816871732473373,
-0.00141045730561018,
0.048051830381155014,
0.057570476084947586,
0.05420580878853798,
0.015810662880539894,
-0.017844220623373985,
0.010099323466420174,
-0.006617405917495489,
0.08284898102283478,
-0.09150862693786621,
0.021099675446748734,
-0.05620046705007553,
-0.08368818461894989,
-0.036802615970373154,
-0.11602462083101273,
0.016523415222764015,
-0.049353230744600296,
0.05897226557135582,
0.07074207812547684,
-0.018404746428132057,
0.1012243926525116,
0.03325319290161133,
0.044759947806596756,
-0.0475328154861927,
-0.039724431931972504,
-0.06983125954866409,
-0.024811288341879845,
-0.02351854369044304,
-0.019708743318915367,
0.008470322005450726,
-0.06563565135002136,
-0.0458303764462471,
-0.0010929652489721775,
0.027803068980574608,
-0.08036668598651886,
0.018094109371304512,
-0.02100527100265026,
0.004769751336425543,
-0.026825593784451485,
0.03285633400082588,
0.01956343464553356,
0.019193632528185844,
0.008805717341601849,
-0.09062400460243225,
-0.02715550735592842,
-0.08774426579475403,
0.0243375226855278,
0.0025864660274237394,
0.0025786426849663258,
0.12498435378074646,
-0.05943181738257408,
0.03200977295637131,
-0.0645306259393692,
-0.06941846013069153,
-0.051303308457136154,
-0.05825989693403244,
0.059948261827230453,
-0.0013098653871566057,
-0.034786731004714966,
-0.013685345649719238,
0.12239804118871689,
0.07120367884635925,
0.01582675240933895,
0.03290031850337982,
-0.039364635944366455,
-0.09322167932987213,
-0.02658015862107277,
0.04412386193871498,
0.01196979358792305,
0.027642423287034035,
-0.008210470899939537,
-0.03450952470302582,
0.08851084858179092,
-0.016258303076028824,
-0.02204737439751625,
-0.05111555755138397,
-0.03728009760379791,
-0.021545737981796265,
-0.01884228177368641,
-0.044516511261463165,
0.042171329259872437,
0.049634527415037155,
0.051786791533231735,
0.05461745336651802,
-0.0012219332857057452,
0.05755912512540817,
0.047205038368701935,
0.016393328085541725,
-0.0005121935391798615,
-0.04973848536610603,
-0.055521637201309204,
-0.013024960644543171,
0.11367230862379074,
0.045615438371896744,
-0.04835907742381096,
0.08915376663208008,
0.04169879108667374,
-0.007899414747953415,
-0.024719418957829475,
0.09846998751163483,
-0.0027598196174949408,
0.01286016684025526,
0.012009871192276478,
0.01052892580628395,
-0.029374049976468086,
0.009637366980314255,
0.028970446437597275,
-0.013382968492805958,
0.026343924924731255,
-0.008656129240989685,
-0.04027299955487251,
-0.011850266717374325,
-0.025009850040078163,
-0.13509829342365265,
0.02120075561106205,
0.03679865971207619,
0.077369324862957,
-0.06378630548715591,
0.004222465679049492,
0.07644175738096237,
0.021237891167402267,
0.12119665741920471,
0.051417842507362366,
-0.09043494611978531,
-0.04621312767267227,
-0.05771580711007118,
-0.028953412547707558,
-0.0748106837272644,
0.016842687502503395,
-0.041262853890657425,
0.04084128886461258,
0.021131888031959534,
0.033692214637994766,
0.007063997443765402,
-0.06962086260318756,
0.04506662115454674,
0.006294150836765766,
-0.09341114014387131,
-0.009125081822276115,
0.0113593228161335,
-0.048644889146089554,
0.060843780636787415,
-4.883423037862232e-33,
0.020245954394340515,
0.00426507368683815,
-0.03079037368297577,
-0.003444345435127616,
-0.03342348337173462,
-0.04833376407623291,
0.05834157392382622,
0.031904205679893494,
0.07233221083879471,
0.09224969893693924,
0.016908051446080208,
0.01930960826575756,
-0.05499280244112015,
0.02269701659679413,
-0.028683792799711227,
-0.03358583524823189,
-0.04401306062936783,
-0.08544814586639404,
-0.02384585328400135,
-0.039510659873485565,
0.015226680785417557,
0.006434714421629906,
-0.011890104040503502,
-0.005370829254388809,
-0.02222171053290367,
0.020535411313176155,
0.07274098694324493,
-0.0650179386138916,
-0.028428208082914352,
0.02285311557352543,
-0.03726501017808914,
-0.04574206843972206,
0.0573449432849884,
0.040624018758535385,
-0.051360126584768295,
0.02746501751244068,
-0.016390936449170113,
0.002459933515638113,
0.03412126377224922,
0.07306656986474991,
0.013610606081783772,
0.11150075495243073,
0.029816381633281708,
0.0012175312731415033,
0.06017021834850311,
0.026657210662961006,
0.011458381079137325,
0.08122260123491287,
-0.002387851011008024,
0.04946858808398247,
-0.04767661541700363,
-0.10601481795310974,
0.02567368559539318,
-0.008974926546216011,
0.0840003490447998,
0.007353303488343954,
-0.004785526543855667,
-0.09411164373159409,
0.05331461504101753,
-0.03046911023557186,
-0.06615190953016281,
-0.08092884719371796,
-0.025404643267393112,
-0.008081378415226936,
-0.0682854950428009,
-0.09688537567853928,
-0.08503589034080505,
-0.08148816972970963,
-0.013027633540332317,
-0.0509318970143795,
0.09873776137828827,
-0.004749355837702751,
-0.04691926762461662,
-0.039322689175605774,
0.0028784573078155518,
-0.08484822511672974,
0.007616596296429634,
0.027016792446374893,
0.04721982404589653,
0.011301782913506031,
-0.06787726283073425,
0.0008940410916693509,
0.005637123249471188,
0.08504506200551987,
-0.00781948957592249,
-0.005503448657691479,
0.10043922811746597,
0.07852096110582352,
0.10774223506450653,
0.07553751766681671,
0.007624847814440727,
0.02025340311229229,
0.011664542369544506,
-0.005045237950980663,
-0.002743582706898451,
6.808519703691529e-32,
-0.036044634878635406,
0.06989707052707672,
-0.002759946510195732,
0.037039611488580704,
0.013231727294623852,
-0.05121518671512604,
0.05564865469932556,
-0.021064355969429016,
0.02126442641019821,
-0.05413135886192322,
0.09432923793792725,
0.08701568841934204,
-0.08425681293010712,
0.09569448977708817,
-0.006894002202898264,
0.009096951223909855,
0.007777726743370295,
-0.051853444427251816,
-0.007738946471363306,
-0.06603562831878662,
-0.07591787725687027,
-0.015662619844079018,
0.008844711817800999,
0.03608229011297226,
0.02253551408648491,
0.004251408390700817,
-0.019444046542048454,
-0.09979811310768127,
-0.02671467326581478,
-0.03318069875240326,
-0.023930877447128296,
-0.021609792485833168,
0.0010104754474014044,
-0.06005154177546501,
0.0350402295589447,
-0.020393436774611473,
-0.0830078274011612,
0.019477542489767075,
-0.004521805793046951,
0.054567717015743256,
0.04443059116601944,
-0.03819117322564125,
-0.043315332382917404,
-0.014506809413433075,
-0.03697403147816658,
0.017607709392905235,
0.04704321548342705,
0.014735812321305275,
-0.0006455820403061807,
-0.07367566227912903,
-0.049496546387672424,
-0.0007881476194597781,
0.014368678443133831,
-0.09274572134017944,
-0.011545585468411446,
-0.031097810715436935,
0.003942204639315605,
0.02589271403849125,
-0.01804659329354763,
0.044050995260477066,
-0.0830347090959549,
0.008652166463434696,
-0.03222215175628662,
0.04104433208703995
] |
82
Chapter 4
1 a On the same axes sketch the gr
aphs of y = x2(x β 2) and y = 2x β x2.
b By solving a suitable equa
tion find the points of intersection of the two graphs.
2 a On the same axes sketch the curv
es with equations y = 6 __ x and y = 1 + x.
b The curves intersect at the points
A and B . Find the coordinates of A and B .
c The curve C with equa
tion y = x2 + px + q, where p and q are integers, passes through A and B .
Find the values of p and q .
d Add C to y
our sketch.
3 The diagram sho
ws a sketch of the curve y = f(x). y
x O2
BA (3, 4)
y = 2
The point B (0, 0) lies on the curve and the point A (3, 4)
is a maximum point. The line y = 2 is an asymptote.
Sketch the following and in each case give the
coordinates of the new positions of A and B and
state the equation of the asymptote:
a f(2x) b 1 _ 2 f(x) c f(x) β 2
d f(x + 3) e f(x β
3) f f(x) + 1
4 The diagram sho
ws the curve with equation
y = 5 + 2x β x2 and the line with equation y = 2.
The curve and the line intersect at the points
A and B .
Find the x -coordinates of A and B . (4 marks)
5 f(x
) = x2(x β 1)(x β 3).
a Sketch the gra
ph of y = f(x). (2 marks)
b On the same axes, dr
aw the line y = 2 β x. (2 marks)
c State the number of
real solutions to the equation x2(x β 1)(x β 3) = 2 β x. (1 mark)
d Write down the coor
dinates of the point where the graph with equation
y = f(x) + 2 crosses the y-axis. (1 mark)
6 The figure shows a sk
etch of the curve with
equation y = f(x).
On separate axes sketch the curves with equations:a
y =
f(β x) (2 marks)
b y =
βf(x) (2 marks)
Mark on each sketch the
x-coordinate of any point,
or points, where the curve touches or crosses the x -axis.P
Ey
x Oy = 2 B A
y = 5 + 2x β x2
E/P
Ey
x O β2 2Mixed exercise 4
|
[
-0.01696573570370674,
0.07196259498596191,
-0.022713154554367065,
0.024965256452560425,
-0.01977226324379444,
0.07208871096372604,
0.008239323273301125,
0.01169667113572359,
-0.07628785818815231,
-0.006508714985102415,
0.00936982873827219,
-0.03800100088119507,
-0.012277021072804928,
0.01382342167198658,
-0.018131040036678314,
0.01012389175593853,
-0.06476754695177078,
-0.027715882286429405,
-0.037311527878046036,
-0.050002001225948334,
-0.018189547583460808,
-0.03124728612601757,
-0.04140578955411911,
-0.05298466607928276,
0.061993345618247986,
-0.0908636823296547,
0.029962090775370598,
-0.035966139286756516,
0.004085923545062542,
-0.035231813788414,
0.11181996762752533,
0.02308644726872444,
-0.01333682332187891,
0.03167126700282097,
0.058025896549224854,
0.05925263836979866,
0.10281265527009964,
0.035737864673137665,
0.02280084230005741,
-0.06878352910280228,
-0.06019311025738716,
-0.058338578790426254,
-0.07152166962623596,
-0.01598075032234192,
0.07044057548046112,
-0.02341901883482933,
-0.046586498618125916,
-0.01811198703944683,
0.040296830236911774,
-0.008345418609678745,
0.024066997691988945,
-0.0027423144783824682,
-0.03983209282159805,
-0.00921513233333826,
-0.002345440909266472,
0.025822889059782028,
-0.050667233765125275,
-0.03109334222972393,
-0.005544615443795919,
0.12022759020328522,
-0.00004335862467996776,
0.030732542276382446,
-0.0187729150056839,
0.06434816867113113,
-0.02910877764225006,
0.03932597115635872,
0.06496786326169968,
0.014312822371721268,
0.005037212744355202,
0.07843703031539917,
-0.038471709936857224,
0.03631960228085518,
-0.04590170457959175,
-0.06531234830617905,
0.015202527865767479,
-0.04938288405537605,
-0.03256654366850853,
-0.041115280240774155,
-0.017231784760951996,
-0.03067612461745739,
-0.04184304550290108,
-0.019394826143980026,
0.05348912253975868,
0.049923405051231384,
-0.021900692954659462,
-0.003222949802875519,
0.023565270006656647,
-0.04645314812660217,
0.059387072920799255,
0.011383438482880592,
0.05959298089146614,
-0.017122449353337288,
-0.04592645913362503,
-0.06451615691184998,
0.022573575377464294,
-0.12509307265281677,
0.057639673352241516,
-0.05399663746356964,
0.011706126853823662,
0.09965550154447556,
0.03793148323893547,
0.014594021253287792,
-0.012172355316579342,
0.10710571706295013,
0.04036417230963707,
0.02649405412375927,
-0.01954522170126438,
-0.0772949606180191,
0.025620583444833755,
-0.032438866794109344,
-0.004677262622863054,
-0.09282899647951126,
0.020993413403630257,
0.062197599560022354,
0.107785664498806,
-0.01653021201491356,
0.06854372471570969,
-0.041120074689388275,
-0.0034247965086251497,
-0.07422134280204773,
0.07936163991689682,
-0.10129678249359131,
0.03371179848909378,
-0.013538457453250885,
-0.04600943624973297,
-0.04548267647624016,
-0.04022785648703575,
-0.029303347691893578,
0.03248997777700424,
-0.062272779643535614,
-0.012662691064178944,
-0.05879559367895126,
-0.007073964457958937,
-0.03556787595152855,
-0.018251748755574226,
-0.002385578118264675,
-0.04915345087647438,
0.06256043165922165,
-0.02739677019417286,
-0.08221878856420517,
0.08178107440471649,
0.08285390585660934,
0.00843378622084856,
-0.037654977291822433,
0.019685976207256317,
0.011065812781453133,
-0.09372322261333466,
0.0052267578430473804,
0.04296638071537018,
-0.016554614529013634,
0.026078233495354652,
0.00552942231297493,
0.03352030739188194,
0.13501866161823273,
0.027744127437472343,
0.0028308217879384756,
-0.0029017606284469366,
-0.016361819580197334,
0.0057548098266124725,
-0.0362829864025116,
-0.10195708274841309,
0.04516609013080597,
-0.10490137338638306,
0.0493515282869339,
0.06798108667135239,
-0.04299521818757057,
-0.004450809210538864,
0.09215737134218216,
0.02515118010342121,
0.021068399772047997,
-0.013218740001320839,
0.053660809993743896,
-0.038279447704553604,
0.033827826380729675,
0.03217398375272751,
-0.043262772262096405,
0.0952782928943634,
0.04817407950758934,
-0.03126849606633186,
0.009576570242643356,
0.010154134593904018,
-0.03629940003156662,
0.01487435307353735,
-0.027716465294361115,
0.02417985163629055,
-0.07022059708833694,
0.017130598425865173,
0.04986773803830147,
-0.002420481527224183,
0.08755598962306976,
-0.027725106105208397,
-0.0771738588809967,
-0.017755698412656784,
0.13523124158382416,
-0.0968664214015007,
0.010022818110883236,
0.0618676133453846,
0.04285574331879616,
-0.12244394421577454,
0.0032900210935622454,
0.024189643561840057,
0.030205221846699715,
0.04295935109257698,
-0.05983605608344078,
0.046655427664518356,
-0.002206079661846161,
-0.06345457583665848,
0.04638279974460602,
-0.046210236847400665,
-0.019994650036096573,
0.060297735035419464,
0.02786155790090561,
-0.03605449199676514,
-0.050276871770620346,
-0.005279793404042721,
-0.0837910994887352,
0.02831936813890934,
-0.06129561364650726,
-0.14012908935546875,
0.028114978224039078,
-0.022081872448325157,
-0.025840485468506813,
0.06357929855585098,
2.012099190432279e-33,
-0.01825857348740101,
-0.006057209335267544,
-0.03127536550164223,
-0.03436203673481941,
-0.0625998005270958,
-0.030493583530187607,
0.10460928827524185,
-0.05197058618068695,
0.10556989163160324,
0.07510534673929214,
0.06554335355758667,
0.010628698393702507,
-0.03907119855284691,
0.07516169548034668,
-0.011483763344585896,
0.009056607261300087,
-0.045415595173835754,
-0.043002642691135406,
-0.019827967509627342,
-0.03360132500529289,
-0.005421206820756197,
-0.029252296313643456,
-0.005856062285602093,
-0.07746700942516327,
0.05203858017921448,
0.06625165790319443,
0.07579155266284943,
-0.12969990074634552,
-0.08106637001037598,
0.03188510239124298,
-0.05111224949359894,
-0.05109792202711105,
0.05164245516061783,
-0.01654146797955036,
-0.04229319840669632,
-0.021248357370495796,
0.009404796175658703,
-0.043277841061353683,
0.03552921116352081,
-0.06853421777486801,
0.03079506941139698,
0.05577250197529793,
0.050990015268325806,
0.06013251096010208,
0.014348870143294334,
0.0487825833261013,
-0.012813800014555454,
0.12404631078243256,
-0.0219862163066864,
0.048289187252521515,
0.015251632779836655,
-0.065278559923172,
0.07336925715208054,
0.006867376621812582,
0.13951687514781952,
-0.014028158970177174,
0.06612657010555267,
-0.03206194192171097,
0.013788647018373013,
-0.08727158606052399,
-0.0024175827857106924,
-0.031690411269664764,
-0.012052719481289387,
0.0715901106595993,
0.02949892356991768,
-0.041745975613594055,
-0.07816685736179352,
-0.05466308444738388,
0.012037727981805801,
0.01931072026491165,
0.03783288970589638,
0.017927279695868492,
-0.0137793542817235,
-0.0845704972743988,
-0.0379612073302269,
0.008154836483299732,
-0.048278965055942535,
0.047709278762340546,
0.06124043092131615,
-0.07719796895980835,
-0.04578465223312378,
0.04376298561692238,
-0.00014983947039581835,
-0.01678190752863884,
-0.026837484911084175,
0.053170908242464066,
0.06095840781927109,
0.027122657746076584,
0.05137944594025612,
-0.020284973084926605,
0.00046458045835606754,
0.024994131177663803,
-0.04537421092391014,
-0.018771925941109657,
0.10249001532793045,
7.998395640148952e-32,
-0.057841431349515915,
-0.041659802198410034,
-0.025677695870399475,
-0.004107610322535038,
0.03143126890063286,
0.05815063416957855,
0.049355920404195786,
0.03663664311170578,
-0.022876692935824394,
-0.025985678657889366,
0.00574328051880002,
0.052697546780109406,
-0.09663122147321701,
0.0408497154712677,
-0.026220232248306274,
0.038552235811948776,
-0.02443615160882473,
0.01884981244802475,
-0.030839109793305397,
-0.058018799871206284,
-0.0661504715681076,
-0.020420202985405922,
-0.03136861324310303,
0.06454556435346603,
-0.022091612219810486,
0.016677072271704674,
-0.04737798124551773,
-0.039194244891405106,
0.019831979647278786,
-0.016498208045959473,
-0.0035937472712248564,
-0.042444150894880295,
-0.07375585287809372,
-0.004118173848837614,
0.09346485137939453,
0.019601086154580116,
-0.09318061918020248,
0.11297371238470078,
0.01445193961262703,
-0.010537593625485897,
-0.06796109676361084,
-0.06244347244501114,
-0.0859338566660881,
0.0260008554905653,
0.013452179729938507,
-0.005029548425227404,
0.06671498715877533,
-0.04292427375912666,
0.051009651273489,
-0.03891107812523842,
-0.07472363114356995,
0.04579343646764755,
-0.012508085928857327,
0.030726183205842972,
0.047076720744371414,
-0.08327571302652359,
-0.06889382749795914,
0.021140290424227715,
0.013852348551154137,
-0.023479100316762924,
-0.08194667100906372,
0.06350594758987427,
-0.05613031983375549,
-0.013788523152470589
] |
83Graphs and transformations
7 The diagram sho
ws the graph of the quadratic function f(x).
x O13
(2, β1)y
y = f(x)
The graph meets the x-axis at (1, 0) and (3, 0) and the
minimum point is (2, β1).
a Find the equation of the gr
aph in the form
y = ax2 + bx = c (2 marks)
b On separate ax
es, sketch the graphs of
i y =
f(x + 2) ii y =
(2x). (2 marks)
c On each graph la
bel the coordinates of the
points at which the graph meets the x-axis and
label the coordinates of the minimum point.
8 f(x
) = (x β 1)(x β 2)(x + 1).
a State the coordina
tes of the point at which the graph y = f(x ) intersects the y -axis. (1 mark)
b The graph of
y = af(x) intersects the y -axis at (0, β 4). Find the value of a . (1 mark)
c The graph of
y = f(x + b) passes through the origin. Find three possible values of b . (3 marks)
9 The point P(4, 3) lies on a curv
e y = f(x).
a State the coordina
tes of the point to which P is transformed on the curve with equation:
i y =
f(3x) ii 1 _ 2 y = f(x) iii y = f(x β 5) iv βy = f(x) v 2( y + 2) = f(x)
b P is transf
ormed to point (2, 3). Write down two possible transformations of f(x).
c P is transf
ormed to point (8, 6). Write down a possible transformation of f(x) if
i f(x
) is translated only
ii f(x
) is stretched only.
10 The curve C1 has equation y = β a __ x2 where a is a positive constant. The curve C2 has the
equation y = x2 (3x + b) where b is a positive constant.
a Sketch C1 and C2 on the same set of axes, showing clearly the coordinates of any
point where the curves touch or cross the axes. (4 marks)
b Using your sketch sta
te, giving reasons, the number of solutions to the equation
x 4 (3x + b) + a = 0. (2 marks)
11 a Factorise completel
y x3 β 6x2 + 9x. (2 marks)
b Sketch the curve of
y = x3 β 6x2 + 9x showing clearly the coordinates of the
points where the curve touches or crosses the axes. (4 marks)
c The point with coordinates (
β4, 0) lies on the curve with equation
y = (x β k)3 β 6(x β k)2 + 9(x β k) where k is a constant.
Find the two possible values of k. (3 marks)
12 f(x
) = x(x β 2)2
Sketch on separate axes the graphs of:
a y =
f(x) (2 marks)
b y =
f(x + 3) (2 marks)
Show on each sketch the coor
dinates of the points where each graph crosses or meets the axes.E/P
E/P
P
E/P
E/P
E
|
[
0.00649389810860157,
0.0698540061712265,
-0.018366096541285515,
-0.10235852748155594,
-0.03687293082475662,
0.06967819482088089,
-0.022405508905649185,
-0.019884124398231506,
-0.05680349841713905,
-0.04473491758108139,
0.000046465018385788426,
-0.02134426310658455,
-0.023704620078206062,
-0.03490864858031273,
-0.05686081200838089,
-0.007920393720269203,
-0.0015226922696456313,
-0.03127157315611839,
-0.015254716388881207,
-0.09327228367328644,
0.041427113115787506,
-0.09336024522781372,
-0.01830012910068035,
-0.1203174740076065,
0.040257763117551804,
-0.13355007767677307,
-0.028055883944034576,
-0.029022101312875748,
-0.06626564264297485,
-0.020671645179390907,
0.00027513393433764577,
0.016386359930038452,
0.06441696733236313,
0.02668989822268486,
0.11855528503656387,
-0.044656816869974136,
0.06977951526641846,
0.004182877484709024,
0.0876900851726532,
-0.021141089498996735,
-0.027314061298966408,
0.04922318831086159,
0.011061719618737698,
-0.032809894531965256,
0.06304211914539337,
-0.015246648341417313,
-0.0651167780160904,
0.03133906424045563,
0.017977621406316757,
-0.025527318939566612,
0.018057696521282196,
0.06924878805875778,
-0.01158554945141077,
0.04544011503458023,
0.05503830686211586,
0.009467875584959984,
0.010779039934277534,
0.05436955764889717,
0.02464381605386734,
0.013481895439326763,
-0.01354260928928852,
0.015571382828056812,
-0.033131975680589676,
0.07402393221855164,
-0.0035825290251523256,
0.09132498502731323,
0.06483794748783112,
-0.042919911444187164,
-0.06337828189134598,
0.06954843550920486,
-0.09935028105974197,
0.0817677304148674,
-0.0467677004635334,
-0.04811578989028931,
0.03739366680383682,
-0.026194358244538307,
0.02085108496248722,
0.019319523125886917,
0.03347114101052284,
-0.05957358330488205,
-0.025067510083317757,
-0.014580599963665009,
0.06841333955526352,
0.055993203073740005,
0.041567228734493256,
-0.0032889368012547493,
0.01288610603660345,
-0.04762793704867363,
-0.043471116572618484,
-0.020032698288559914,
-0.016600284725427628,
0.08878449350595474,
0.001317356713116169,
-0.022059781476855278,
-0.004405227955430746,
-0.1680135577917099,
-0.016613800078630447,
-0.018660906702280045,
0.04019756242632866,
0.09918875992298126,
-0.0504520982503891,
0.04546206444501877,
0.005386847537010908,
0.07202330976724625,
-0.012688789516687393,
-0.0275993924587965,
0.009934189729392529,
-0.0773400291800499,
0.011008378118276596,
0.0003009748470503837,
-0.10989196598529816,
-0.040570154786109924,
-0.0073737469501793385,
0.06165569648146629,
0.08259511739015579,
-0.040042564272880554,
-0.02415435202419758,
-0.0006036476115696132,
0.037455279380083084,
-0.033343859016895294,
0.009755810722708702,
-0.01340056024491787,
0.08140960335731506,
0.011326069943606853,
-0.07626951485872269,
-0.03034329228103161,
-0.0782114788889885,
0.03973649442195892,
0.013192003592848778,
-0.02738737314939499,
0.030905157327651978,
-0.08782771229743958,
0.01797964796423912,
-0.059982456266880035,
0.03188565745949745,
-0.0659516230225563,
-0.09456818550825119,
0.06624360382556915,
-0.039848338812589645,
-0.03546271100640297,
0.05932076647877693,
0.07658100873231888,
0.03421067073941231,
0.03923967480659485,
0.027202019467949867,
-0.03542531654238701,
-0.08056356012821198,
0.0036306926049292088,
-0.030164940282702446,
-0.01777685433626175,
-0.0900665745139122,
-0.007808813359588385,
-0.018047817051410675,
0.10161673277616501,
0.05065104737877846,
-0.06727270036935806,
-0.0780056044459343,
-0.06489945203065872,
-0.05760476365685463,
-0.05860789120197296,
-0.04517456516623497,
0.04823087528347969,
-0.06276551634073257,
0.039337433874607086,
0.08347491919994354,
-0.030945852398872375,
0.04755297675728798,
0.0665426179766655,
0.029464270919561386,
0.05132567137479782,
0.006673627067357302,
0.027832845225930214,
0.01758798398077488,
0.11290954053401947,
0.010793513618409634,
-0.01579681783914566,
0.12434930354356766,
0.020698465406894684,
-0.06012628600001335,
0.06911906599998474,
0.06893378496170044,
-0.00656979251652956,
0.026377413421869278,
0.01036175899207592,
-0.022549761459231377,
-0.05409261956810951,
0.013748986646533012,
0.009634530171751976,
-0.048702314496040344,
0.024376822635531425,
-0.003297485178336501,
-0.014408654533326626,
-0.012499925680458546,
0.04796870797872543,
-0.12325073778629303,
-0.029324403032660484,
0.011488931253552437,
0.05512199550867081,
-0.10600744187831879,
-0.01701465994119644,
0.09531546384096146,
0.04134206473827362,
0.06272546947002411,
0.05202523246407509,
0.0010047265095636249,
-0.014710335060954094,
-0.12701211869716644,
-0.05115346983075142,
-0.07102634757757187,
0.02437254600226879,
0.007674023974686861,
-0.012330293655395508,
-0.02607092820107937,
-0.0002734167210292071,
-0.01039733923971653,
-0.1269102394580841,
-0.02635432407259941,
-0.05687625706195831,
-0.13030174374580383,
-0.025925127789378166,
0.007345275487750769,
0.005917869973927736,
0.020500456914305687,
-2.018344371513007e-33,
0.02924133464694023,
0.03234904259443283,
-0.028273994103074074,
-0.021132366731762886,
-0.03716098144650459,
-0.07373303174972534,
0.07445000857114792,
-0.01204424537718296,
0.062302909791469574,
0.1440601348876953,
0.07340873032808304,
-0.01990644447505474,
-0.006087415385991335,
-0.02959243394434452,
-0.06753204762935638,
-0.02486294135451317,
-0.0070136128924787045,
-0.012168597429990768,
-0.028612764552235603,
-0.05077457055449486,
0.039897020906209946,
0.048506882041692734,
0.03053559549152851,
0.015024637803435326,
-0.011945427395403385,
0.02920186147093773,
0.09387033432722092,
-0.06695330888032913,
-0.016954215243458748,
0.04780139401555061,
-0.053426533937454224,
0.011646646074950695,
0.005114346276968718,
0.01718214713037014,
-0.030307887122035027,
0.0023930049501359463,
0.0135091133415699,
-0.06406491249799728,
0.055822473019361496,
0.001006735023111105,
0.07919865846633911,
0.04756617918610573,
0.01694902405142784,
-0.001422631205059588,
0.06604058295488358,
0.035782698541879654,
0.02588259056210518,
0.04024336114525795,
0.017830554395914078,
0.06096348911523819,
0.011826765723526478,
-0.0897243469953537,
0.0041930824518203735,
-0.013492336496710777,
0.10647501051425934,
0.007787936832755804,
0.00019816264102701098,
-0.07359476387500763,
-0.002432876266539097,
-0.07258839160203934,
-0.01547207497060299,
-0.02913244627416134,
-0.05560443922877312,
0.0748770460486412,
-0.04492385312914848,
-0.1030515506863594,
-0.021486550569534302,
-0.047179076820611954,
0.0002624989429023117,
-0.00842826534062624,
0.041826747357845306,
0.03548111021518707,
0.039232946932315826,
-0.0750131905078888,
-0.037564948201179504,
-0.01572078838944435,
-0.041898686438798904,
0.04548675939440727,
0.009579523466527462,
-0.07731647789478302,
-0.04050249233841896,
-0.0026376775931566954,
0.01419773232191801,
0.02391403168439865,
0.03604689985513687,
0.0011805640533566475,
0.07780764251947403,
0.041572876274585724,
0.11147347092628479,
0.015569797717034817,
0.04173315316438675,
0.007494180463254452,
-0.01958329975605011,
-0.02687647007405758,
0.021530237048864365,
7.271796721029978e-32,
-0.02376691624522209,
0.039000365883111954,
-0.02792425826191902,
0.006882620509713888,
0.0631820559501648,
0.012836987152695656,
0.041994187980890274,
-0.0579800121486187,
-0.038291335105895996,
-0.045798469334840775,
0.04940733313560486,
0.14075089991092682,
-0.05292532593011856,
0.06871983408927917,
-0.026027264073491096,
-0.009493441320955753,
-0.024437015876173973,
0.005729201249778271,
-0.017926204949617386,
-0.04149876534938812,
-0.05300695449113846,
-0.005173036362975836,
-0.042594268918037415,
0.10004989057779312,
-0.0721748098731041,
-0.021934784948825836,
-0.039004452526569366,
-0.09446481615304947,
0.022352954372763634,
-0.03099614568054676,
-0.04367074370384216,
0.07043168693780899,
-0.02254575490951538,
-0.03714723885059357,
0.054703980684280396,
-0.006907064933329821,
-0.06163410097360611,
0.03340201452374458,
-0.014775498770177364,
0.03236076980829239,
-0.024805817753076553,
-0.03808889910578728,
-0.015053587965667248,
0.010285935364663601,
-0.036744993180036545,
0.04642312973737717,
0.05796685442328453,
-0.06251966953277588,
0.027383899316191673,
-0.060202471911907196,
0.034702759236097336,
0.0015733273467049003,
0.08403395861387253,
-0.06615373492240906,
-0.06786200404167175,
-0.06433660537004471,
-0.00466235401108861,
0.02013787068426609,
0.010923714376986027,
0.016992520540952682,
-0.016952836886048317,
0.05698772892355919,
-0.02517494186758995,
-0.007594537455588579
] |
84
Chapter 4
13 Given tha
t f(x) = 1 __ x , x β 0,
a Sketch the gra
ph of y = f(x) β 2 and state the equations of the asymptotes. (3 marks)
b Find the coordinates of
the point where the curve y = f(x) β 2 cuts a coordinate
axis. (2 marks)
c Sketch the gra
ph of y = f(x + 3). (2 marks)
d State the equations of
the asymptotes and the coordinates of the point where
the curve cuts a coordinate axis. (2 marks)E
The point R (6, β 4) l ies on the curve with equation y = f( x). State the coordinates
that point R is transformed to on the curve with equation y = f( x + c ) β d .Challenge
1 If p is a r oot of the function f( x), then the graph of y = f(x) touches or crosses the x-axis at
the point (p, 0).
2 The graphs of y = k __ x and y = k __ x2 , where k is a real constant, have asymptotes at x = 0 and y = 0.
3 The x-coordinate(s) at the points of intersection of the curves with equations y = f(x) and
y = g(x) are the solution(s) to the equation f( x) = g( x).
4 The graph of y = f(x) + a is a translation of the graph y = f(x) by the vector ( 0 a ) .
5 The graph of y = f(x + a) is a translation of the graph y = f(x) by the vector ( βa 0 ) .
6 When you translat
e a function, any asymptotes are also translated.
7 The graph of y = af(x) is a stretch of the graph y = f(x) by a scale factor of a in the vertical
direction.
8 The graph of y = f(ax) is a stretch of the graph y = f( x) by a scale factor of 1 __ a in the horizontal
direction.
9 The graph of y = βf( x) is a reflection of the graph of y = f(x) in the x-axis.
10 The graph of y = f(β x) is a reflection of the graph of y = f(x) in the y-axis.Summary of key points
|
[
0.005548852030187845,
0.11908348649740219,
0.001617048867046833,
0.014689534902572632,
-0.009964574128389359,
-0.019769061356782913,
0.0358993262052536,
0.10491347312927246,
-0.018443966284394264,
0.0034839999862015247,
0.07910178601741791,
-0.06592093408107758,
-0.05153549462556839,
0.013913234695792198,
-0.026045240461826324,
-0.06177949532866478,
-0.08349082618951797,
-0.018455786630511284,
-0.010413069278001785,
-0.027566615492105484,
0.014335487969219685,
0.046293970197439194,
-0.00894168484956026,
-0.022035110741853714,
0.06539039313793182,
-0.10062602907419205,
-0.03325473144650459,
-0.0728820338845253,
-0.009429003112018108,
-0.06518832594156265,
0.024636201560497284,
0.011488445103168488,
0.0014302694471552968,
-0.0004975875490345061,
0.07031066715717316,
0.01294636819511652,
0.03966343775391579,
-0.02806512825191021,
0.07251310348510742,
-0.00900453794747591,
0.0056058494374156,
0.057565443217754364,
-0.05816502869129181,
-0.030940404161810875,
0.14190636575222015,
-0.02597106620669365,
-0.07645648717880249,
-0.01311611570417881,
0.007091313134878874,
0.018675612285733223,
-0.048446156084537506,
0.06875333935022354,
-0.08976081013679504,
0.07622350752353668,
-0.0038112588226795197,
0.012251888401806355,
0.04077920690178871,
0.04673052579164505,
0.011353516951203346,
0.08518736809492111,
0.017463451251387596,
0.07110632210969925,
-0.039137035608291626,
0.03838373348116875,
-0.009669274091720581,
0.05115458369255066,
0.012062495574355125,
-0.06380557268857956,
0.04684069752693176,
0.10866934061050415,
-0.1507103145122528,
-0.0015853835502639413,
-0.01342515368014574,
-0.09041725099086761,
0.07314001768827438,
-0.017380939796566963,
0.0010053908918052912,
0.029975222423672676,
-0.05379663035273552,
-0.02542945183813572,
-0.01831378974020481,
0.024243665859103203,
0.08216328173875809,
0.03325739875435829,
0.010487040504813194,
-0.0024813972413539886,
0.008795910514891148,
-0.027409853413701057,
0.004040316212922335,
-0.035848040133714676,
0.013604188337922096,
0.049046240746974945,
-0.03503280133008957,
0.0005580068100243807,
0.030808862298727036,
-0.05772903561592102,
-0.02101757936179638,
-0.04744500294327736,
0.05479278042912483,
0.10670804232358932,
-0.07830437272787094,
0.03151436150074005,
-0.0026577734388411045,
0.036068156361579895,
-0.02636478841304779,
-0.032036490738391876,
0.05850338935852051,
-0.08662385493516922,
0.011765970848500729,
-0.061891429126262665,
-0.010694360360503197,
-0.040045365691185,
0.03934238478541374,
0.05189768224954605,
0.10529026389122009,
-0.030787155032157898,
0.01497979462146759,
-0.0542517714202404,
-0.021150868386030197,
-0.05768870189785957,
0.053262900561094284,
-0.07333630323410034,
-0.005462340544909239,
-0.024462521076202393,
-0.05736960470676422,
0.010855683125555515,
-0.06241929531097412,
0.003491281531751156,
0.030912643298506737,
0.021300183609128,
0.09473946690559387,
-0.11027568578720093,
-0.012667655013501644,
-0.01560221891850233,
-0.07072105258703232,
-0.04486508294939995,
-0.0693851038813591,
0.05853298678994179,
0.018819058313965797,
-0.02872154675424099,
0.034453537315130234,
0.08839883655309677,
-0.08865343779325485,
0.05320873484015465,
0.07240547239780426,
0.0023019223008304834,
0.0060902065597474575,
-0.005462737288326025,
0.009954163804650307,
-0.027072779834270477,
-0.005961144808679819,
0.011060497723519802,
-0.005418367218226194,
0.04642815515398979,
0.10254082828760147,
0.038818903267383575,
-0.060797207057476044,
-0.04119044169783592,
-0.07569564878940582,
0.0011154274689033628,
-0.04670143499970436,
0.043529026210308075,
-0.01159887295216322,
0.014208689332008362,
0.060628004372119904,
-0.007725376635789871,
0.07444553822278976,
0.045679595321416855,
-0.04189812391996384,
0.0878789871931076,
0.041737258434295654,
-0.015963567420840263,
0.03392885625362396,
0.06651098281145096,
0.0009860377758741379,
-0.004306707065552473,
0.12053344398736954,
0.07437489926815033,
-0.0044561042450368404,
0.029865948483347893,
0.03021932765841484,
-0.05386530980467796,
0.0540301650762558,
-0.0027871711645275354,
-0.045135073363780975,
-0.0864655077457428,
-0.030604196712374687,
0.06096033751964569,
0.021004565060138702,
0.052717313170433044,
-0.0041549052111804485,
-0.04435137286782265,
-0.02813607268035412,
0.05868792161345482,
-0.07374905794858932,
-0.0713464766740799,
-0.0017591712530702353,
0.01814696565270424,
-0.03028956986963749,
0.017836133018136024,
-0.0023066133726388216,
0.009501401335000992,
-0.023954523727297783,
0.02097475528717041,
-0.035640597343444824,
0.04513755440711975,
-0.07565057277679443,
0.031125273555517197,
-0.06504318863153458,
-0.11701833456754684,
0.03849590942263603,
-0.0017990126507356763,
-0.07881607115268707,
-0.006584624759852886,
0.016872022300958633,
-0.05731244385242462,
0.04120059311389923,
-0.08798430860042572,
-0.12904495000839233,
0.05400008708238602,
0.02774316631257534,
-0.016154004260897636,
0.01294801663607359,
7.035808295515535e-33,
0.03137928992509842,
0.034479204565286636,
-0.03437446802854538,
-0.02763807401061058,
-0.03853057697415352,
-0.10306347906589508,
0.0834323912858963,
0.046910375356674194,
0.05519266054034233,
0.04785962402820587,
0.07722488790750504,
0.04645445570349693,
-0.029333319514989853,
-0.0033307888079434633,
0.04521641880273819,
-0.045022912323474884,
0.032065827399492264,
0.07184354215860367,
-0.02765454538166523,
-0.00306219351477921,
-0.030548911541700363,
0.013875598087906837,
0.03499671816825867,
-0.0030808954034000635,
-0.007466591894626617,
0.009760686196386814,
-0.009717773646116257,
-0.0992635041475296,
-0.10300970822572708,
0.05292773246765137,
-0.06441584974527359,
-0.017428163439035416,
-0.02936888299882412,
-0.03975986689329147,
-0.049560222774744034,
-0.03422878310084343,
-0.03332878276705742,
-0.018616266548633575,
0.01014676783233881,
0.029816001653671265,
0.014278206042945385,
0.021213581785559654,
0.08077194541692734,
-0.031009385362267494,
-0.016878029331564903,
0.024698903784155846,
0.06301194429397583,
0.07378140836954117,
-0.05317874997854233,
0.041503556072711945,
-0.021250857040286064,
-0.08107901364564896,
0.019033309072256088,
0.030841946601867676,
0.10967262089252472,
0.05674426257610321,
0.020329387858510017,
-0.031105173751711845,
-0.013837633654475212,
-0.1282157003879547,
-0.029648426920175552,
-0.022090714424848557,
-0.04763982817530632,
-0.01561175286769867,
-0.05590497702360153,
-0.07173390686511993,
-0.08194547891616821,
-0.05910254269838333,
0.036148566752672195,
0.020066265016794205,
0.025908615440130234,
0.010062657296657562,
0.011308497749269009,
-0.06830870360136032,
-0.06813625991344452,
-0.030364476144313812,
-0.10276521742343903,
-0.046807125210762024,
0.05027338117361069,
-0.07711640745401382,
-0.010871405713260174,
0.026693809777498245,
-0.04537395387887955,
0.06759530305862427,
-0.041495125740766525,
-0.006164376623928547,
0.021230395883321762,
0.0675099790096283,
0.09001464396715164,
-0.026721376925706863,
-0.01735357567667961,
0.012259083800017834,
-0.009007973596453667,
-0.014441238716244698,
0.04901449754834175,
7.501446587131198e-32,
-0.04448629170656204,
-0.020867574959993362,
0.02873925492167473,
0.060940567404031754,
0.07687897980213165,
0.04741733521223068,
0.041828688234090805,
-0.07658777385950089,
-0.03091522492468357,
-0.10440968722105026,
0.02885906957089901,
0.048516277223825455,
-0.08149775862693787,
0.05909363925457001,
-0.08325400948524475,
-0.007171081844717264,
0.013894339092075825,
0.04364557936787605,
-0.009223978966474533,
-0.018200740218162537,
-0.050020329654216766,
0.04417767748236656,
-0.09352651238441467,
0.03258037194609642,
0.03769480064511299,
-0.01743527129292488,
-0.007815640419721603,
-0.09266122430562973,
-0.008962288498878479,
-0.05424867942929268,
0.034054066985845566,
-0.0382031686604023,
0.08210776001214981,
-0.04022723436355591,
0.028084682300686836,
-0.0055125062353909016,
-0.04013727977871895,
-0.010257003828883171,
-0.0005816894117742777,
0.02889222651720047,
-0.03127316012978554,
0.0073763709515333176,
-0.037374284118413925,
-0.005260374862700701,
-0.03578667342662811,
-0.003506293287500739,
0.018266228958964348,
-0.030279511585831642,
0.09050892293453217,
0.05097818002104759,
0.02297055348753929,
0.11259568482637405,
0.024187350645661354,
-0.07455607503652573,
0.0019237082451581955,
0.039059195667505264,
-0.004458884708583355,
-0.022708678618073463,
-0.09855298697948456,
0.030772840604186058,
-0.026469001546502113,
0.1370929777622223,
-0.04408349469304085,
0.03963598981499672
] |
Review exercise
851
1 a Write down the value of 8 1 _ 3 . (1 mark)
b Find the value of
8 β 2 _ 3 . (2 marks)
β Section 1.4
2 a Find the value of 12 5 4 _ 3 . (2 marks)
b Simplify 24x2 Γ· 18 x 4 _ 3 . (2 marks)
β Sections 1.1, 1.4
3 a Express ββ―___ 80 in the form a ββ―__
5 ,
where a is an integer. (2 marks)
b Express (4 β
ββ―__
5 )2 in the form b + c ββ―__
5 ,
where b and c are integers. (2 marks)
β Section 1.5
4 a Expand and simplify
(4 +
ββ―__
3 )(4 β ββ―__
3 ). (2 marks)
b Express 26 ______ 4 + ββ―__
3 in the form a + b ββ―__
3 ,
where a and b are integers. (3 marks)
β Sections 1.5, 1.6
5 Here are three numbers:
1 β
β __
k , 2 + 5 β __
k and 2 β __
k
Given tha
t k is a positive integer, find:
a the mean of the three
n
umbers. (2 marks)
b the range of the thr
ee
numbers. (1 mark)
β Section 1.5
6 Given that y = 1 ___ 25 x 4 , express each of the
following in the form kxn, where k and n
are constants.
a yβ1 (1 mark)
b 5 y 1 _ 2 (1 mark)
β Section 1.4E
E
E
E/p
E7 Find the area of this tr apezium in cm2.
Give your answer in the form a + b β __
2 ,
where a and b are integers to be
found. (4 marks)
β Section 1.5
(5 + 3 2) cm3 + 2 cm
2 2 cm
8 Given that p = 3 β 2 β __
2 and q = 2 β β __
2 ,
find the value of p + q _____ p β q .
Give y
our answer in the form m + n β __
2 ,
where m and n are rational numbers to be
found. (4 marks)
β Sections 1.5, 1.6
9 a Factorise the expression x
2 β 10x +16. (1 mark)
b Hence, or otherwise, solv
e the equation
82y β 10(8y) + 16 = 0. (2 marks)
β Sections 1.3, 2.1
10 x2 β 8x β 29 (x + a)2 + b, where a and b
are constants.
a Find the value of
a and the value
of b. (2 marks)
b Hence, or otherwise, sho
w that the
roots of x2 β 8x β 29 = 0 are c Β± d ββ―__
5 ,
where c and d are integers. (3 marks)
β Sections 2.1, 2.2E/p
E
E/p
E
|
[
0.0707051083445549,
0.08695950359106064,
-0.004012358840554953,
-0.05049767345190048,
0.01238033827394247,
0.12074267119169235,
-0.012086556293070316,
-0.00391438277438283,
-0.0718291699886322,
-0.0012865080498158932,
0.01211061142385006,
-0.09725109487771988,
0.027754759415984154,
-0.024009214714169502,
-0.0017249444499611855,
-0.0464085154235363,
-0.036924369633197784,
0.059266943484544754,
-0.08911590278148651,
-0.004200151655822992,
0.07396434247493744,
-0.05872536078095436,
-0.0059335967525839806,
0.02715851366519928,
0.09299325197935104,
-0.053293947130441666,
-0.07000410556793213,
-0.044830322265625,
-0.045877885073423386,
-0.07675151526927948,
0.04869697988033295,
0.013005804270505905,
0.14114698767662048,
-0.038370199501514435,
-0.026811674237251282,
-0.0013247184688225389,
-0.0019887336529791355,
0.028726018965244293,
-0.011166810989379883,
-0.024433355778455734,
-0.0626077651977539,
0.0013973413733765483,
0.007916436530649662,
0.04046576842665672,
0.01315094530582428,
-0.05191420018672943,
0.0409221351146698,
0.01819656789302826,
0.04809866100549698,
-0.007454471196979284,
0.005635540932416916,
0.03603582829236984,
-0.06781864166259766,
0.017277279868721962,
0.03176940977573395,
-0.06610820442438126,
0.012979457154870033,
-0.03620663285255432,
-0.06766986101865768,
0.02174576371908188,
-0.019781362265348434,
0.009304533712565899,
0.0026467922143638134,
0.002641103696078062,
-0.0375232994556427,
0.07764872163534164,
0.024552643299102783,
-0.06836439669132233,
0.04448015242815018,
0.02346515841782093,
-0.0554322712123394,
0.06532319635152817,
0.011896220035851002,
-0.05239289999008179,
0.0443725548684597,
-0.008317110128700733,
-0.16789332032203674,
0.00011550044291652739,
-0.005390547215938568,
-0.060470100492239,
-0.03338673710823059,
-0.02851513773202896,
0.0680197924375534,
-0.035834427922964096,
0.009964781813323498,
0.003499689046293497,
0.013328689150512218,
0.09661708027124405,
0.04994821920990944,
-0.05324147269129753,
0.08185138553380966,
0.0033405935391783714,
-0.020111462101340294,
0.024543603882193565,
0.0783572643995285,
0.016443738713860512,
-0.0298455897718668,
-0.08714959770441055,
0.022065995261073112,
0.08788267523050308,
0.046666469424963,
-0.015486150048673153,
-0.009544221684336662,
-0.01930665224790573,
-0.10629045218229294,
0.023838359862565994,
-0.007183799520134926,
-0.04510989785194397,
0.07121523469686508,
-0.1180553287267685,
-0.07476826012134552,
0.030674757435917854,
0.003905755467712879,
0.008311162702739239,
-0.007227268069982529,
0.031180614605545998,
0.09998564422130585,
0.017050765454769135,
-0.0287946667522192,
0.005949995014816523,
0.052648723125457764,
-0.041915636509656906,
0.117420494556427,
-0.05174098163843155,
-0.00927889347076416,
-0.051545239984989166,
0.03703109920024872,
0.009333058260381222,
-0.038993727415800095,
-0.0482349768280983,
0.037706926465034485,
-0.04476980119943619,
-0.04969331622123718,
-0.012004547752439976,
-0.0648299902677536,
-0.11659584194421768,
-0.02132245898246765,
0.08148883283138275,
-0.00012659034109674394,
-0.0843525379896164,
0.08059433102607727,
0.05862627923488617,
-0.007170720957219601,
-0.05027611926198006,
-0.04576489329338074,
0.06032446026802063,
-0.015829917043447495,
-0.03117837943136692,
0.019671248272061348,
0.02550882287323475,
-0.03481539711356163,
0.032443877309560776,
0.00990096665918827,
0.08753234148025513,
0.002046414418146014,
-0.06226416304707527,
0.019225360825657845,
-0.023829041048884392,
-0.002172544365748763,
0.030856192111968994,
-0.08651931583881378,
0.03939739242196083,
0.0016196424840018153,
0.04966389760375023,
0.11053585261106491,
0.022693302482366562,
0.03935560584068298,
0.036713797599077225,
-0.01182086206972599,
-0.06370195001363754,
0.03391622379422188,
-0.03466019406914711,
0.05663277208805084,
0.08128964155912399,
0.02763913758099079,
-0.07896064966917038,
0.12119323760271072,
0.0018323652911931276,
-0.01887124963104725,
-0.013507887721061707,
0.09637758880853653,
0.007707905489951372,
0.002906948793679476,
0.025982722640037537,
0.007956573739647865,
0.0054731955751776695,
0.02696058712899685,
0.04022044315934181,
0.018433136865496635,
0.016160380095243454,
-0.013594957999885082,
-0.07324394583702087,
-0.09207389503717422,
0.007717890664935112,
-0.039074160158634186,
-0.0525553897023201,
-0.010749584063887596,
-0.030949823558330536,
-0.12473519146442413,
0.040926653891801834,
0.005146848037838936,
0.019551998004317284,
0.0867171660065651,
-0.029312167316675186,
0.025763481855392456,
-0.023043183609843254,
-0.00964417029172182,
-0.05421599745750427,
0.08494306355714798,
0.04467279464006424,
0.0004947782726958394,
-0.03915939852595329,
-0.06608816981315613,
-0.12104838341474533,
0.04261113703250885,
0.03459103778004646,
0.0010544315446168184,
-0.04027592018246651,
-0.03378484770655632,
-0.02724437601864338,
-0.00746871717274189,
-0.1108986884355545,
0.06149822473526001,
6.61238274072187e-33,
-0.1054893508553505,
0.06632282584905624,
-0.1206897646188736,
-0.030638422816991806,
-0.03909694403409958,
-0.00021106787608005106,
0.054285816848278046,
-0.03839978575706482,
0.04335048422217369,
0.060407690703868866,
0.008961673825979233,
-0.008755385875701904,
0.05593961849808693,
0.04714295268058777,
-0.08268367499113083,
-0.03838139772415161,
-0.130987748503685,
0.05024341866374016,
0.07919103652238846,
-0.056785523891448975,
-0.02474728412926197,
0.0023297490552067757,
0.0057465373538434505,
0.06531641632318497,
-0.010624924674630165,
0.032033372670412064,
0.0590183399617672,
-0.08997795730829239,
0.01429719664156437,
0.056763432919979095,
-0.02087077684700489,
-0.050657160580158234,
0.05277621001005173,
0.0019162100506946445,
-0.04275483638048172,
-0.06285656988620758,
0.04083411395549774,
0.010496268048882484,
0.05508117005228996,
0.0344448983669281,
0.11710517108440399,
0.009475107304751873,
0.01870156265795231,
0.004550708923488855,
0.02005738392472267,
-0.02869267389178276,
0.06067292392253876,
0.05928459390997887,
-0.0005043658311478794,
-0.0176861509680748,
-0.030426902696490288,
0.01399365533143282,
-0.05601543188095093,
-0.00016784077160991728,
0.04701452702283859,
-0.03133682161569595,
0.03812011703848839,
-0.010312850587069988,
0.08322380483150482,
0.0007452519494108856,
-0.07210583984851837,
-0.01622661016881466,
0.0221718680113554,
0.015495491214096546,
0.0016271136701107025,
-0.0560590997338295,
-0.07193167507648468,
0.07534654438495636,
-0.007724976632744074,
0.014689713716506958,
0.01796780154109001,
0.019918039441108704,
-0.08023063093423843,
-0.05730057880282402,
0.03738703206181526,
0.01877032220363617,
0.06068304926156998,
0.055527858436107635,
-0.07403720170259476,
-0.02979445271193981,
-0.08448547124862671,
-0.028958357870578766,
-0.0030038373079150915,
-0.05553768202662468,
-0.0667521134018898,
-0.021981025114655495,
0.12344997376203537,
0.07440470904111862,
-0.04503175988793373,
0.027209162712097168,
-0.03034505993127823,
-0.019702469930052757,
0.02924024872481823,
-0.04007946327328682,
-0.01960047520697117,
8.844466515360924e-32,
-0.018581299111247063,
-0.01127565372735262,
-0.008193831890821457,
-0.000661384838167578,
0.016057314351201057,
0.03649770841002464,
-0.022313015535473824,
0.05188992992043495,
0.08477551490068436,
-0.017594289034605026,
0.10739828646183014,
-0.030721666291356087,
-0.03771086037158966,
0.060826387256383896,
-0.07378478348255157,
-0.03832605481147766,
0.01388145238161087,
-0.008355550467967987,
-0.05044243112206459,
0.008353889919817448,
0.027180349454283714,
0.017803329974412918,
-0.003411374520510435,
0.047431349754333496,
0.05293600261211395,
0.027083158493041992,
-0.10523845255374908,
0.024483732879161835,
-0.0009376640664413571,
-0.044575031846761703,
0.01703587733209133,
0.03522159531712532,
-0.0367666594684124,
-0.025099774822592735,
-0.03995019569993019,
0.030486080795526505,
0.034235041588544846,
0.05585091561079025,
-0.0025511595886200666,
0.09168846160173416,
-0.07540443539619446,
-0.09814441949129105,
-0.01050176564604044,
-0.029144305735826492,
0.09489460289478302,
-0.05772608518600464,
-0.05148377642035484,
0.024455348029732704,
-0.04446837306022644,
-0.05065309628844261,
-0.08262190967798233,
0.009863419458270073,
0.00711795873939991,
0.0031813308596611023,
0.02837243117392063,
-0.048823390156030655,
-0.021563135087490082,
-0.06961777061223984,
-0.04873838648200035,
0.018467292189598083,
0.006912630051374435,
0.05630222335457802,
-0.010655484162271023,
0.057055868208408356
] |
86
Review exercise 1
11 The functions f and g are defined as
f(
x) = x(x β 2) and g(x) = x + 5, x β β .
Given tha
t f(a) = g(a) and a > 0,
find the value of a to three significant figures.
(3 marks)
β Sections 2.1, 2.3
12 An athlete launches a shot put from shoulder height. The height of the shot put, in metr
es, above the ground
tΒ seconds after launch, can be modelled by the following function:
h(t) = 1.7 + 10t β 5t
2 t > 0
a Give the ph
ysical meaning of the
constant term 1.7 in the context of the
model.
b Use the model to calcula
te how many
seconds after launch the shot put hits the ground.
c
Rearrange h(
t) into the form
AΒ βΒ B(tΒ βΒ C)2 and give the values of the
constants A, B and C.
d Using your answ
er to part c or
otherwise, find the maximum height of the shot put, and the time at which this maximum height is reached.
β Section 2.6
13 Given that f(x) = x2 β 6x + 18, x > 0,
a express f(
x) in the form (x β a)2 + b,
where a and b are integers. (2 marks)
The curve C
with equation y = f(x),
x > 0, meets the y-axis at P and has a minimum point at Q.
b
Sketch the gra
ph of C, showing the
coordinates of P and Q. (3 marks)
The line y
= 41 meets C at the point R.
c Find the x-coor
dinate of R, giving
your answer in the form p + q ββ―__
2 ,
where p and q are integers. (2 marks)
β Sections 2.2, 2.4
14 The function h(x) = x2 + 2 β __
2 x + k has
equal roots.
a Find the value of
k. (1 mark)
b Sketch the gra
ph of y = h(x), clearly
labelling any intersections with the
coordinate axes. (3 marks)
β Sections 1.5, 2.4, 2.5E/p
p
E/p
E15 The function g(x) is defined as
g(
x) = x9 β 7x6 β 8x3, x β β .
a Write g(x ) in the f
orm x3(x3 + a)(x3 + b),
where a and b are integers. (1 mark)
b Hence find the three roots
of
g(x). (1 mark)
β Section 2.3
16 Given that
x2 + 10x + 36 (x + a)2 + b,
where a and b are constants,
a find the value of
a and the value
of b. (2 marks)
b Hence show that the equa
tion
x2 + 10x + 36 = 0 has no
real roots. (2 marks)
The equation x2 + 10x + k = 0 has equal
roots.c
Find the value of
k. (2 marks)
d For this va
lue of k, sketch the graph
of y = x2 + 10x + k, showing the
coordinates of any points at which
theΒ graph meets the coordinate axes.
(3 marks)
β Sections 2.2, 2.4, 2.5
17 Given that x2 + 2x + 3 (x + a)2 + b,
a find the value of
the constants
a and b (2 marks)
b Sketch the gra
ph of y = x2 + 2x + 3,
indicating clearly the coordinates of any intersections with the coordinate axes.
(3 marks)
c Find the value of
the discriminant of
x2 + 2x + 3. Explain how the sign of
the discriminant relates to your sketch in part b.
(2 marks)
The equation x2 + kx + 3 = 0, where k is a
constant, has no real roots.
d Find the set of possible v
alues
of k, giving your answer in surd
form. (2 marks)
β Section 2.2, 2.4, 2.5E/p
E/p
E/p
|
[
0.015314262360334396,
0.11579277366399765,
0.06447824835777283,
-0.059130195528268814,
-0.035781342536211014,
0.02937139943242073,
0.024608595296740532,
0.03117590956389904,
-0.031325217336416245,
0.03970177844166756,
0.059722110629081726,
-0.03183305263519287,
-0.05115228146314621,
0.04436951130628586,
-0.02598971500992775,
-0.025063736364245415,
0.018505828455090523,
-0.026985250413417816,
-0.06742679327726364,
-0.05282562971115112,
-0.017092786729335785,
0.0020691263489425182,
0.07149980962276459,
-0.05359949171543121,
-0.011897364631295204,
-0.049358151853084564,
-0.058813270181417465,
-0.027117885649204254,
-0.04697456210851669,
-0.011751745827496052,
-0.10324868559837341,
-0.07879148423671722,
0.02663411572575569,
-0.020866544917225838,
-0.03752555325627327,
0.041104137897491455,
-0.05909966304898262,
0.0480327233672142,
0.041230667382478714,
-0.005510509014129639,
-0.004314708989113569,
-0.055949583649635315,
-0.002142858924344182,
-0.06577138602733612,
0.0737091451883316,
0.07357905060052872,
0.004874274600297213,
0.036354195326566696,
-0.02656937949359417,
0.047202255576848984,
-0.018051862716674805,
0.07907012104988098,
-0.09919475764036179,
-0.006158596370369196,
-0.023260798305273056,
-0.05710068717598915,
0.04659261554479599,
-0.06527301669120789,
0.025419708341360092,
0.04501606896519661,
-0.054656073451042175,
0.06826740503311157,
0.005819341633468866,
0.021497955545783043,
0.03567029908299446,
0.05436820909380913,
-0.044944897294044495,
-0.09696820378303528,
0.044269412755966187,
0.11036030203104019,
-0.0867692306637764,
0.07676584273576736,
-0.029067300260066986,
-0.09784730523824692,
-0.0001269236672669649,
-0.04263560473918915,
-0.030640285462141037,
-0.022556010633707047,
-0.05590025708079338,
0.02341311052441597,
0.005465212278068066,
-0.053585827350616455,
-0.030103743076324463,
0.08574934303760529,
0.007244102191179991,
0.061945561319589615,
0.07820035517215729,
0.10489989817142487,
-0.03636323660612106,
0.02628064528107643,
-0.010178196243941784,
0.016517922282218933,
-0.014282655902206898,
-0.048578858375549316,
-0.046819057315588,
0.009497400373220444,
0.019565680995583534,
-0.0648886188864708,
-0.018557967618107796,
0.13998094201087952,
0.02851772867143154,
-0.010989687405526638,
0.058628592640161514,
0.0646958276629448,
0.03775377944111824,
-0.02008148841559887,
0.022091897204518318,
-0.009311678819358349,
-0.06537112593650818,
-0.0037338093388825655,
0.05529339984059334,
0.024576406925916672,
-0.007366901263594627,
0.08553139865398407,
0.04724578931927681,
-0.037374455481767654,
0.04505099728703499,
-0.01989760808646679,
0.036459244787693024,
-0.11015399545431137,
0.07137666642665863,
-0.049973271787166595,
0.020912302657961845,
-0.10537557303905487,
-0.08260507881641388,
-0.052041176706552505,
0.026584787294268608,
0.0189583208411932,
-0.06445902585983276,
-0.04598028212785721,
0.03390990570187569,
-0.009395741857588291,
-0.041859451681375504,
-0.06689132750034332,
-0.06611844152212143,
-0.06915193796157837,
-0.01800556480884552,
0.04371769353747368,
0.0005699643515981734,
-0.018794972449541092,
0.017535774037241936,
0.01757480390369892,
0.06975565105676651,
0.017321299761533737,
0.030702386051416397,
0.058771759271621704,
-0.00989497173577547,
0.013645058497786522,
0.04523877054452896,
-0.039529141038656235,
0.026408281177282333,
0.030094264075160027,
0.04584677889943123,
0.1460220366716385,
-0.009335549548268318,
-0.030867457389831543,
-0.09220384061336517,
-0.08056968450546265,
-0.06269939243793488,
-0.047243114560842514,
-0.11934943497180939,
0.0005934836808592081,
-0.06029490381479263,
-0.07094144821166992,
0.056147050112485886,
-0.008077315054833889,
0.08216950297355652,
0.005357114132493734,
-0.05084752291440964,
0.01475655846297741,
0.00735063711181283,
-0.07373294979333878,
-0.04146223142743111,
0.019222483038902283,
0.040356505662202835,
0.0066634006798267365,
0.04826151579618454,
0.011035967618227005,
-0.018786584958434105,
-0.032310061156749725,
0.08157491683959961,
-0.10044325888156891,
0.02457364648580551,
0.03554992377758026,
0.03980835899710655,
-0.03687293827533722,
-0.025140227749943733,
0.029568396508693695,
0.06898277252912521,
0.035690922290086746,
0.025240426883101463,
-0.03278549388051033,
0.004170448984950781,
0.0369456522166729,
-0.0830039232969284,
-0.006521514151245356,
0.0032792517449706793,
0.1114678680896759,
0.014050581492483616,
0.048294685781002045,
0.07421379536390305,
-0.027164869010448456,
0.038686603307724,
0.05589822307229042,
0.032629359513521194,
-0.03881075233221054,
-0.019218243658542633,
-0.00776089821010828,
0.0224089827388525,
-0.019298477098345757,
-0.09288238734006882,
-0.043462369590997696,
-0.0925285667181015,
0.005926464684307575,
0.06744575500488281,
0.011652360670268536,
0.00624007498845458,
-0.013980682939291,
-0.13514584302902222,
0.058148596435785294,
-0.05307692661881447,
0.02830100990831852,
0.020540907979011536,
5.904652122238326e-33,
-0.02273366041481495,
0.04996070638298988,
-0.04020993411540985,
-0.02779567986726761,
0.06476490944623947,
-0.030665017664432526,
0.03825576603412628,
-0.0350613072514534,
0.0701899379491806,
0.09057990461587906,
0.04945722222328186,
0.02590247429907322,
-0.05366210639476776,
-0.044340040534734726,
0.015347158536314964,
-0.08116379380226135,
0.004974523559212685,
-0.016753170639276505,
-0.14872770011425018,
0.025134995579719543,
0.12400471419095993,
0.01639821194112301,
0.021509042009711266,
-0.011370746418833733,
-0.026014648377895355,
0.004718757700175047,
0.049078021198511124,
-0.05789439007639885,
-0.0012991997646167874,
-0.033747341483831406,
-0.010335474275052547,
0.01853804849088192,
0.08395586162805557,
0.1104620024561882,
-0.016675367951393127,
-0.05080045387148857,
0.09331407397985458,
0.03470601513981819,
0.01996578276157379,
-0.03660762682557106,
0.015346490778028965,
0.014000301249325275,
0.043101854622364044,
0.06123398616909981,
0.002997295232489705,
-0.006077783647924662,
-0.020989172160625458,
0.006013606674969196,
-0.03447619453072548,
0.029787935316562653,
-0.003337122732773423,
-0.03752699866890907,
-0.002674586372449994,
-0.002206065459176898,
0.07056307047605515,
-0.03441878780722618,
0.0439462773501873,
-0.04776200279593468,
0.041178323328495026,
0.025507714599370956,
0.018979663029313087,
-0.03785330429673195,
-0.014523602090775967,
0.10052263736724854,
-0.09102863818407059,
-0.0046396986581385136,
-0.07018040120601654,
-0.05994395911693573,
-0.07637479901313782,
0.07282332330942154,
0.003937805537134409,
0.07072791457176208,
0.0874418243765831,
-0.07669781893491745,
-0.09335830062627792,
-0.016604091972112656,
-0.08458198606967926,
-0.02484617568552494,
0.046400286257267,
-0.0207805298268795,
-0.017479535192251205,
-0.047126587480306625,
0.04731728509068489,
0.037582628428936005,
-0.0566081665456295,
0.027475396171212196,
-0.005954962223768234,
0.001752647920511663,
0.025923877954483032,
0.020136384293437004,
0.03871002420783043,
0.10556209832429886,
-0.006901327054947615,
-0.030045025050640106,
-0.02684546448290348,
7.143629045476771e-32,
-0.04884756729006767,
0.08207646757364273,
-0.051622048020362854,
0.05206990987062454,
-0.01989372819662094,
0.0748598501086235,
-0.017561810091137886,
-0.0995379239320755,
0.03333429992198944,
-0.026487169787287712,
0.037813421338796616,
0.018533596768975258,
-0.07531726360321045,
0.006184047088027,
-0.06976206600666046,
-0.08754794299602509,
-0.0376114696264267,
0.003153276862576604,
-0.0599086657166481,
0.008275818079710007,
-0.04062916338443756,
0.00708600040525198,
-0.07892774045467377,
0.10729900002479553,
0.0360541045665741,
-0.036456480622291565,
-0.07099339365959167,
0.019579477608203888,
0.006173888221383095,
-0.035717081278562546,
0.01427086815237999,
0.035258352756500244,
-0.03859460726380348,
-0.031948018819093704,
0.04901540279388428,
0.027428211644291878,
-0.02286618761718273,
0.007299497723579407,
-0.03123178705573082,
0.006539948750287294,
-0.0067948708310723305,
-0.021174954250454903,
0.023536039516329765,
0.04532017558813095,
-0.037000950425863266,
-0.05333438515663147,
0.02015833742916584,
-0.04803701117634773,
0.055676914751529694,
0.04653478413820267,
-0.017021888867020607,
0.08657486736774445,
-0.04325403273105621,
0.020975643768906593,
0.03232723847031593,
0.036605581641197205,
0.02320488728582859,
-0.026214828714728355,
-0.03563787788152695,
-0.08059823513031006,
-0.04669622331857681,
0.0642331913113594,
-0.05666934698820114,
0.056818295270204544
] |
87
Review exercise 1
18 a By eliminating
y from the equations:
y = x β 4,
2x2 β xy = 8,
show that
x2 + 4x β 8 = 0. (2 marks)
b Hence, or otherwise, solv
e the
simultaneous equations:
y = x β 4,2x
2 β xy = 8,
giving your answers in the form a Β± b
ββ―__
3 , where a and b are
integers. (4 marks)
β Section 3.2
19 Find the set of va lues of x for which:
a 3(2x
+ 1) > 5 β 2x, (2 marks)
b 2x2 β 7x + 3 > 0, (3 marks)
c both 3(2
x + 1) > 5 β 2x and
2x2 β 7x + 3 > 0. (1 mark)
β Sections 3.4, 3.5
20 The functions p and q are defined as
p(x
) = β2(x + 1) and q(x) = x2 β 5x + 2,
x β β . Show alge
braically that there is no
value of x for which p(x ) = q(x). (3 marks)
β Sections 2.3, 2.5
21 a Solve the simultaneous equations:
y
+ 2x = 5
2x2 β 3x β y = 16. (5 marks)
b Hence, or otherwise, find the set of
values of x for which:
2x2 β 3x β 16 > 5 β 2x. (2 marks)
β Sections 3.2, 3.5
22 The equation x2 + kx + (k + 3) = 0, where
k is a constant, has different real roots.
a Show that
k2 β 4k β 12 > 0. (2 marks)
b Find the set of possible v
alues of k.
(2 marks)
β Sections 2.5, 3.5
23 Find the set of va lues for which
6 _____ x + 5 < 2, x β β5. (6 marks)
β Section 3.5E
E
E/p
E
E/p
E24 The functions f and g are defined as
f(
x) = 9 β x2 and g(x) = 14 β 6x, x β β .
a On the same set of axes
, sketch the
graphs of y = f(x) and y = g(x). Indicate
clearly the coordinates of anyΒ points where the graphs intersect with each other or the coordinate axes.
(5 marks)
b On your sketch, shade the r
egion that
satisfies the inequalities y > 0 and f(x)Β >Β g(x).
(1 mark)
β Sections 3.2, 3.3, 3.7
25 a Factorise completely x3 β 4x. (1 mark)
b Sketch the curve with equa
tion
y = x3 β 4x, showing the coordinates of
the points where the curve crosses the x-axis.
(2 marks)
c On a separate dia
gram, sketch the
curve with equation
y = (x β 1)3 β 4(x β 1)
showing the coordinates of the pointsΒ where the curve crosses the x-axis.
(2 marks)
β Sections 1.3, 4.1, 4.5
26
O
P(3, β2)y
x 2 4
The figure shows a sketch of the curve with equation y = f(x). The curve crosses the x-axis at the points (2, 0) and (4, 0). The minimum point on the curve is P(3, β2).
In separate diagrams, sketch the curves
with equation
a
y =
βf(x) (2 marks)
b y = f(2
x) (2 marks)
On each diagram, gi
ve the coordinates of
the points at which the curve crosses the
x-axis, and the coordinates of the image of P under the given transformation.
β Sections 4.6, 4.7E
E/p
E
|
[
0.008966542780399323,
0.07969505339860916,
0.008859998546540737,
-0.0068662879057228565,
0.007101622875779867,
0.012006348930299282,
0.012291046790778637,
-0.02132152020931244,
-0.06928626447916031,
0.03507368639111519,
0.08422739803791046,
-0.0928497388958931,
0.06862367689609528,
-0.004826024174690247,
-0.024570206180214882,
-0.04775827378034592,
-0.007351549807935953,
-0.009028511121869087,
-0.06355880200862885,
0.05461300164461136,
0.05307672917842865,
-0.060573920607566833,
-0.048891838639974594,
-0.010353636927902699,
0.1067146584391594,
-0.04373284429311752,
-0.08431561291217804,
-0.03205154463648796,
-0.01802819035947323,
-0.023331156000494957,
0.0722072497010231,
0.050618916749954224,
0.07520664483308792,
0.011988821439445019,
0.06689861416816711,
0.0036530352663248777,
0.01545090414583683,
0.06317295134067535,
0.020813006907701492,
-0.10097850859165192,
-0.05698699504137039,
-0.07226266711950302,
0.0017579242121428251,
-0.044963810592889786,
0.04815037176012993,
-0.040408018976449966,
0.08872638642787933,
-0.01316913589835167,
0.05282389745116234,
-0.040057189762592316,
0.020201193168759346,
0.03635668754577637,
-0.004010007716715336,
0.06923088431358337,
0.00014242291217669845,
-0.03746770694851875,
0.04624234512448311,
-0.018010787665843964,
0.006669399328529835,
0.060407888144254684,
-0.0016167431604117155,
-0.01944134756922722,
-0.004654333461076021,
0.03055817447602749,
0.023503849282860756,
0.019720375537872314,
-0.04338585585355759,
-0.019337302073836327,
0.08540351688861847,
0.01143922284245491,
-0.0323813296854496,
0.09605973213911057,
-0.048031728714704514,
-0.02390499971807003,
0.03490201383829117,
0.007272736635059118,
-0.06687630712985992,
-0.07435891777276993,
0.059635378420352936,
-0.007957211695611477,
0.004964700434356928,
-0.02488475665450096,
0.03931016847491264,
-0.028240634128451347,
0.024587707594037056,
-0.036335621029138565,
-0.003592178924009204,
0.0422181598842144,
0.008795768953859806,
-0.05280943214893341,
0.014008789323270321,
0.01838582567870617,
-0.006948210299015045,
0.01961139403283596,
0.10828983783721924,
-0.10519774258136749,
0.047203004360198975,
-0.07919228821992874,
-0.01485128328204155,
0.09685392677783966,
-0.017293155193328857,
-0.022281020879745483,
-0.018228335306048393,
0.0030315984040498734,
0.02451992966234684,
0.040829263627529144,
0.021613461896777153,
-0.03729415684938431,
0.08133363723754883,
-0.04375011846423149,
-0.03769981861114502,
-0.016353892162442207,
-0.008845677599310875,
0.06262258440256119,
-0.015096377581357956,
0.03189586475491524,
0.09930521994829178,
0.062090691179037094,
-0.017011748626828194,
-0.08649802207946777,
-0.0053949179127812386,
-0.05432987958192825,
0.09392689913511276,
-0.030602559447288513,
-0.046550214290618896,
-0.062447454780340195,
-0.025381524115800858,
0.0024128968361765146,
-0.04301910847425461,
-0.07963637262582779,
0.015783915296196938,
-0.07919459789991379,
-0.031346820294857025,
-0.06060022488236427,
-0.08047167956829071,
-0.08382467180490494,
-0.0884329155087471,
0.0922289490699768,
-0.0005738101899623871,
-0.06346062570810318,
0.031727444380521774,
0.04595131799578667,
0.0499342679977417,
0.00036768504651263356,
-0.027370227500796318,
0.021845189854502678,
0.012121552601456642,
-0.024068910628557205,
0.040150854736566544,
0.03572206199169159,
-0.09112630784511566,
-0.028968967497348785,
-0.011610317975282669,
0.06915295124053955,
-0.03694390133023262,
-0.041227295994758606,
-0.057269494980573654,
-0.0064156269654631615,
-0.0233730748295784,
0.024217922240495682,
-0.1018182784318924,
0.02844177559018135,
0.009991500526666641,
0.03978951275348663,
0.0041769128292799,
0.06136699393391609,
0.023169957101345062,
0.07727628946304321,
0.01262239646166563,
-0.005928018130362034,
0.015405677258968353,
0.04601896181702614,
0.04728484898805618,
0.051309652626514435,
0.025333721190690994,
-0.10132409632205963,
0.08726243674755096,
0.034995097666978836,
-0.0691022276878357,
0.056422941386699677,
0.04552885517477989,
-0.06149476766586304,
-0.04450864717364311,
0.00469611119478941,
0.05549109727144241,
0.017707576975226402,
0.01289958693087101,
0.018425459042191505,
-0.0372617170214653,
0.016548385843634605,
-0.03365078568458557,
-0.06933127343654633,
-0.014506649225950241,
0.04341058433055878,
-0.054727837443351746,
-0.06260563433170319,
0.038873616605997086,
-0.07626376301050186,
-0.1297796219587326,
0.1064758151769638,
0.03242786228656769,
0.03841078281402588,
0.11350156366825104,
-0.06582623720169067,
0.06654303520917892,
0.003503232728689909,
-0.012557393871247768,
0.016036301851272583,
0.09306419640779495,
0.03266804292798042,
0.05696260184049606,
0.06880276650190353,
-0.06330624967813492,
-0.0708969458937645,
-0.026495756581425667,
-0.03609035909175873,
0.042376577854156494,
-0.04782792180776596,
-0.06565188616514206,
-0.016992950811982155,
-0.058161091059446335,
-0.08093500137329102,
0.06632530689239502,
4.984860577992014e-35,
-0.010027439333498478,
0.06019110977649689,
-0.10186640173196793,
-0.03903132304549217,
-0.05010534077882767,
-0.0355132520198822,
0.041020046919584274,
-0.057409267872571945,
0.08952954411506653,
0.03405202925205231,
0.02916898764669895,
-0.031164435669779778,
0.01898682303726673,
0.10181206464767456,
0.011464256793260574,
0.006015438120812178,
-0.1158953607082367,
0.010732277296483517,
0.09033177047967911,
0.0007713476079516113,
-0.03127172961831093,
0.011848601512610912,
-0.04593115299940109,
-0.022807952016592026,
0.044223155826330185,
0.026172403246164322,
0.06062260642647743,
-0.04390115663409233,
-0.045718926936388016,
0.054842621088027954,
-0.012041996233165264,
-0.10457215458154678,
0.029900263994932175,
0.010145684704184532,
0.02583879418671131,
-0.016120463609695435,
-0.03667273372411728,
-0.09099745750427246,
0.0211959145963192,
-0.0623411126434803,
0.05981941893696785,
0.07245281338691711,
0.019859835505485535,
0.03199049085378647,
0.0710163414478302,
0.06388892978429794,
0.02657272294163704,
0.045977748930454254,
-0.016993429511785507,
0.04727834463119507,
-0.016850149258971214,
-0.036033958196640015,
-0.10921148955821991,
0.023263635113835335,
0.07296962291002274,
-0.12394307553768158,
0.03582795336842537,
0.04880102351307869,
0.0056739808060228825,
-0.0024442970752716064,
-0.045295316725969315,
-0.0806250274181366,
0.03312794491648674,
0.014058900997042656,
-0.0024322562385350466,
0.01066659390926361,
-0.03393315151333809,
-0.0003826842294074595,
0.00790372584015131,
0.04245408624410629,
0.02405608631670475,
-0.033126406371593475,
-0.07972454279661179,
-0.04973370581865311,
-0.04951756075024605,
-0.013732362538576126,
-0.03384524956345558,
0.03459445387125015,
0.028276856988668442,
-0.04665515199303627,
-0.12854495644569397,
-0.008159930817782879,
-0.0032202431466430426,
-0.021953457966446877,
-0.10444875806570053,
0.023980528116226196,
0.0744573175907135,
0.04674839228391647,
0.027706170454621315,
-0.007630136329680681,
0.044287145137786865,
-0.0075128753669559956,
0.05411029979586601,
-0.026956908404827118,
0.027590082958340645,
8.163885849087941e-32,
-0.04543222114443779,
0.005707947537302971,
-0.052206963300704956,
-0.010160801000893116,
0.013188144192099571,
-0.016320139169692993,
-0.05541395768523216,
0.09750384092330933,
0.003605710342526436,
-0.05661440268158913,
0.076349176466465,
0.0052175624296069145,
-0.10537684708833694,
0.0652720257639885,
-0.044196728616952896,
-0.0011656847782433033,
-0.0176070686429739,
0.007350846193730831,
-0.04200273007154465,
-0.07682476937770844,
-0.09679315984249115,
0.02910904586315155,
-0.054269615560770035,
0.05256574600934982,
0.09379930794239044,
0.007637700531631708,
-0.1005992516875267,
0.00834929384291172,
0.025931507349014282,
0.04229554161429405,
0.03878770023584366,
-0.04816429316997528,
-0.0484146811068058,
-0.12300930172204971,
0.04611373320221901,
0.0023004922550171614,
-0.008120703510940075,
0.0532015822827816,
-0.004728897474706173,
0.029068538919091225,
-0.0418723039329052,
0.010418171994388103,
-0.008718582801520824,
-0.0032837828621268272,
0.021919643506407738,
0.0018403290305286646,
0.003428442869335413,
-0.048991262912750244,
-0.009725157171487808,
-0.057780731469392776,
-0.0593985877931118,
0.03456917405128479,
0.05164821445941925,
0.06181420758366585,
0.00442889891564846,
-0.054229531437158585,
-0.010671899653971195,
-0.021616721525788307,
0.03929220139980316,
-0.04256686568260193,
-0.06875500082969666,
0.10154394805431366,
-0.07300976663827896,
0.010529372841119766
] |
88
Review exercise 1
27
13
4
Oy
x
The figure shows a sketch of the curve
with equation y = f(x). The curve passes through the points (0, 3) and (4, 0) and touches the x-axis at the point (1, 0).
On separate diagrams, sketch the curves
with equations
a
y = f(
x + 1) (2 marks)
b y =
2f(x) (2 marks)
c y = f ( 1 _ 2 x) (2 marks)
On each diagram, sho
w clearly the
coordinates of all the points where the
curve meets the axes.
β Sections 4.5, 4.6, 4.7
28 Given that f(x) = 1 __ x , x β 0,
a sketch the gra
ph of y = f(x) + 3 and
state the equations of the
asymptotes (2 marks)
b find the coordinates of
the point
where y = f(x) + 3 crosses a coordinate axis.
(2 marks)
β Sections 4.3, 4.5
29 The quartic function t is defined as t(x
) = (x2 β 5x + 2)(x2 β 5x + 4), x β β .
a Find the four roots of
t(x), giving your
answers to 3 significant figures where necessary.
(3 marks)
b Sketch the gra
ph of y = t(x), showing
clearly the coordinates of all the pointsΒ where the curve meets the axes.
(2 marks)
β Sections 4.2, 2.1
30 The point (6, β8) lies on the gr aph of
yΒ =Β f(x). State the coordinates of the point to which P is transformed on the graph with equation:
a
y =
βf(x) (1 mark)
b y =
f(x β 3) (1 mark)
c 2y
= f(x) (1 mark)
β Section 4.7E
E
E
E31 The curve C1 has equation y = β a __ x , where
a is a positive constant.
The curve C2 has equation y = (x β b)2,
where b is a positive constant.a
Sketch C1 and C2 on the same set of
axes. Label any points where either
curve meets the coordinate axes, givingΒ your coordinates in terms of a and b.
(4 marks)
b Using your sketch, sta
te the number of
real solutions to the equation x(x β 5)
2 = 7. (1 mark)
β Sections 4.3, 4.4
32 a Sketch the graph of y = 1 __ x 2 β 4 ,
showing clearl
y the coordinates of
the points where the curve crosses
the coordinate axes and stating theΒ equations of the asymptotes.
(4 marks)
b The curve with y
= 1 _______ (x + k)2 β 4 passes
thr
ough the origin. Find the two
possible values of k. (2 marks)
β Sections 4.1, 4.5, 4.7E/p
E/p
1 a Solve the equation x2Β βΒ 10 xΒ +Β 9Β =Β 0
b Hen
ce, or otherwise, solve the equation
3x β 2(3x β 10) = β 1 β Sections 1.1, 1.3, 2.1
2 A rectangle has an area of 6 cm2 and a perimeter
of 8 ββ―__
2 cm. Find the dimensions of the
re
ctangle, giving your answers as surds in their
simplest form. β Sections 1.5, 2.2
3 Show algebraically that the graphs of
yΒ
= 3x3 + x2 β x and y = 2 x(x β 1)( x + 1) have
only one point of intersection, and find the coordinates of this point.
β Section 3.3
4 The quartic function f( x) = (x2 + x β 20)( x2 + x β 2)
has three roots in common with the function
g(x) = f(x β k), where k is a constant. Find the two
possible values of k . β Sections 4.2, 4.5, 4.7Challenge
|
[
0.03879060968756676,
0.057683419436216354,
-0.027005931362509727,
0.0028135222382843494,
0.012351617217063904,
0.0805431380867958,
0.003618121612817049,
0.01488406676799059,
-0.04734433814883232,
0.018454739823937416,
0.04179245978593826,
-0.0864107683300972,
-0.024212485179305077,
0.0494622141122818,
-0.07968652993440628,
-0.02317138761281967,
-0.07763411104679108,
-0.062059950083494186,
0.03076784871518612,
-0.039638616144657135,
-0.009425719268620014,
-0.03788195922970772,
-0.0645427256822586,
-0.09516147524118423,
-0.01999323070049286,
-0.07241927832365036,
0.002937916899099946,
-0.07934916764497757,
-0.11192311346530914,
-0.07053488492965698,
0.03660154715180397,
0.022319860756397247,
0.0028197907377034426,
0.04805145412683487,
0.04149412363767624,
0.003918641712516546,
0.06191985681653023,
0.01618197374045849,
0.10670001059770584,
0.008061720058321953,
-0.048956505954265594,
-0.03916831314563751,
0.033718280494213104,
-0.0228322334587574,
0.12424539029598236,
-0.0428069531917572,
0.021063227206468582,
-0.010594142600893974,
0.014036349952220917,
0.028012264519929886,
0.027656251564621925,
-0.002071729861199856,
-0.08599455654621124,
-0.03252299502491951,
0.05423698201775551,
0.0956779271364212,
0.01488474477082491,
0.010054233483970165,
-0.02619079314172268,
0.05786609277129173,
0.002138913841918111,
0.035919055342674255,
-0.007996239699423313,
0.02810177020728588,
-0.019882744178175926,
0.08800265938043594,
0.05825922265648842,
-0.001300465315580368,
0.01753373257815838,
0.09136800467967987,
-0.1036202684044838,
-0.0013080800417810678,
-0.016378648579120636,
-0.08205341547727585,
0.017452867701649666,
-0.010531567968428135,
0.012172332964837551,
0.0910680741071701,
-0.0833563283085823,
-0.0686897560954094,
-0.10773154348134995,
-0.0201805979013443,
0.12603561580181122,
0.062354087829589844,
-0.03232505917549133,
0.04146556556224823,
-0.014554946683347225,
-0.09507229924201965,
0.00497622461989522,
0.0024365060962736607,
-0.00016532724839635193,
0.05203936621546745,
-0.0743970200419426,
-0.05343165621161461,
-0.010518716648221016,
-0.12003854662179947,
-0.005648317281156778,
-0.05980209633708,
-0.03308532387018204,
0.1190599873661995,
-0.11747933179140091,
0.044404298067092896,
0.01651168055832386,
0.07669994235038757,
-0.0558754988014698,
0.037075404077768326,
-0.009791314601898193,
-0.047912828624248505,
-0.027447110041975975,
-0.05143610015511513,
0.012194307520985603,
-0.030758880078792572,
-0.025431668385863304,
0.044347286224365234,
0.08593190461397171,
-0.031514763832092285,
0.031793735921382904,
-0.056531716138124466,
-0.013100650161504745,
-0.0468551330268383,
0.05683578550815582,
-0.043721120804548264,
0.04700641706585884,
0.010678998194634914,
-0.04585455358028412,
-0.019601233303546906,
-0.04175631329417229,
-0.01638760231435299,
0.02560502290725708,
0.01852116547524929,
0.0416109524667263,
-0.06817511469125748,
-0.017567329108715057,
-0.04947420582175255,
-0.019964244216680527,
0.027017580345273018,
-0.11423337459564209,
0.11121595650911331,
-0.011026439256966114,
-0.054588042199611664,
0.04023456946015358,
0.04580700397491455,
-0.012682047672569752,
-0.07535658031702042,
0.0611572228372097,
-0.020432379096746445,
-0.04217223450541496,
0.006837534252554178,
0.012244801968336105,
0.027256369590759277,
0.02573966421186924,
-0.0390021838247776,
-0.0136440834030509,
0.08738157153129578,
0.00021310585725586861,
-0.013086766935884953,
-0.04485761374235153,
-0.05819661542773247,
0.009131232276558876,
-0.05469280481338501,
-0.07644260674715042,
0.013034932315349579,
-0.031073585152626038,
0.032854802906513214,
0.04232173040509224,
-0.024590076878666878,
0.0015306257409974933,
0.09894758462905884,
0.03426630049943924,
-0.022187380120158195,
0.03431333974003792,
-0.03284836933016777,
-0.04304467514157295,
0.08299339562654495,
-0.01065852027386427,
0.02896290458738804,
0.08488832414150238,
0.08070492744445801,
0.020508596673607826,
0.014900946989655495,
0.07450776547193527,
-0.04371541738510132,
-0.01884409412741661,
-0.07648268342018127,
-0.033159732818603516,
-0.06988084316253662,
-0.04059656709432602,
0.027155278250575066,
-0.012427106499671936,
0.028636101633310318,
-0.026073100045323372,
-0.036211561411619186,
-0.05165902525186539,
0.06721673160791397,
-0.08803368359804153,
0.025053661316633224,
0.09647753834724426,
0.06996780633926392,
-0.04024209454655647,
0.04953594133257866,
-0.026170803233981133,
0.011656928807497025,
0.06902025640010834,
0.08215314894914627,
-0.0458473339676857,
-0.02192862331867218,
-0.004544925410300493,
0.036866866052150726,
-0.016856011003255844,
-0.03323729708790779,
-0.061282120645046234,
0.00026313753915019333,
-0.06471768021583557,
-0.012457199394702911,
0.005507797002792358,
-0.08716244995594025,
0.04601983726024628,
-0.0036402540281414986,
-0.09705370664596558,
0.01167603861540556,
-0.01812008023262024,
0.019919106736779213,
0.04854649677872658,
1.6861118317462115e-33,
-0.0248711034655571,
0.10080765932798386,
0.0073120808228850365,
0.007059447932988405,
-0.027377663180232048,
-0.039814289659261703,
0.07460713386535645,
0.06369994580745697,
0.05277438834309578,
0.08974961936473846,
0.04097527638077736,
-0.012357749976217747,
-0.05456734076142311,
-0.013162939809262753,
0.029458170756697655,
-0.014066859148442745,
-0.004462561570107937,
-0.002378455363214016,
-0.05409688502550125,
-0.07672159373760223,
0.04871390014886856,
0.04201047867536545,
0.06303130835294724,
-0.026125796139240265,
-0.05417356640100479,
0.05321985110640526,
0.06136958301067352,
-0.10121514648199081,
-0.06857713311910629,
0.054951343685388565,
-0.03597484529018402,
0.001313293818384409,
0.0425480455160141,
-0.01002942118793726,
0.011796249076724052,
0.030849818140268326,
0.008576199412345886,
-0.028203638270497322,
0.09955189377069473,
0.027847344055771828,
0.049612339586019516,
0.08993741124868393,
0.054323531687259674,
0.0019505080999806523,
0.006051521748304367,
0.04621125012636185,
-0.013076038099825382,
0.04862705245614052,
0.006608001887798309,
0.016461113467812538,
-0.06294657289981842,
-0.09573563188314438,
0.06109670549631119,
-0.010123908519744873,
0.044789791107177734,
0.042860567569732666,
0.018417097628116608,
-0.14104028046131134,
-0.010177269577980042,
-0.10842812061309814,
-0.0022721460554748774,
-0.0063034724444150925,
-0.09718944132328033,
0.07468179613351822,
0.01804210990667343,
-0.07570397108793259,
-0.1067599281668663,
-0.05004042014479637,
0.002860196866095066,
-0.0055104829370975494,
0.014097939245402813,
0.026868147775530815,
-0.05108645558357239,
-0.0721031129360199,
-0.0075706844218075275,
-0.020675310865044594,
-0.03844689205288887,
0.006375971715897322,
0.04600679501891136,
-0.05422881618142128,
0.021461186930537224,
0.11113445460796356,
-0.037358399480581284,
0.06278737634420395,
-0.016716845333576202,
0.03632132709026337,
0.058385465294122696,
0.04185739532113075,
0.08223722875118256,
0.034393735229969025,
0.03411196172237396,
0.01924309879541397,
-0.05396168306469917,
-0.05533063784241676,
-0.013665981590747833,
8.357476250483337e-32,
-0.12241000682115555,
0.0005162435118108988,
0.03473542258143425,
0.015361754223704338,
0.006681185215711594,
0.018870949745178223,
0.05785109102725983,
-0.02742677368223667,
0.015174498781561852,
-0.047311801463365555,
0.050334226340055466,
0.026570595800876617,
-0.07315083593130112,
0.07867300510406494,
0.009332549758255482,
-0.0035644688177853823,
0.023243946954607964,
-0.021108733490109444,
-0.060102641582489014,
0.0016165929846465588,
0.002996788825839758,
-0.02624598890542984,
-0.026654260233044624,
0.024776402860879898,
0.0044893305748701096,
0.05941648408770561,
0.018886420875787735,
-0.0737064853310585,
-0.0628829374909401,
-0.05395198613405228,
0.0033554090186953545,
-0.03819292411208153,
0.04328586906194687,
0.024539252743124962,
0.06226792559027672,
0.01391880214214325,
-0.019916780292987823,
0.02158466912806034,
-0.05638926848769188,
-0.007834172807633877,
-0.018277788534760475,
-0.048322390764951706,
0.03891594335436821,
0.0013095729518681765,
-0.042421773076057434,
-0.0492437370121479,
-0.008109897375106812,
-0.007896491326391697,
0.010177713818848133,
0.0295582078397274,
-0.06245240196585655,
0.026839086785912514,
0.04261099547147751,
-0.030009683221578598,
0.0213920995593071,
-0.0027099300641566515,
0.01345055177807808,
-0.03841998800635338,
-0.02090814895927906,
0.09818798303604126,
-0.12341894209384918,
0.10755957663059235,
-0.013683199882507324,
-0.05519302561879158
] |
89
Straight line graphs
After completing this unit you should be able to:
β Calculat
e the gradient of a line joining a pair of points β pages 90 β 91
β Understand the link between the equation o f a line, and its gradient
and intercept β pages 91 β 93
β Find the equation of a line given (i) the gr adient and one point on
the line or (ii) two points on the line β pages 93 β 95
β Find the point of intersection f or a pair of straight lines
β pages 95 β 96
β Know and use the rules for parallel and perpendicular gradients
β pages 97 β 100
β Solve length and area problems on coordinate grids β pages 100 β 103
β Use straight line graphs to construct mathematical models
β pages 103 β 108Objectives
1 Find the point of intersection o f the
following pairs of lines.a
y =
4x + 7 and 3y = 2x β 1
b y =
5x β 1 and 3x + 7y = 11
c 2x
β 5y = β1 and 5x β 7y = 14
β GC SE Mathematics
2 Simplify each of the following:
a ββ―___ 80 b ββ―____ 200 c ββ―____ 125
β Sec tion 1.5
3 Make y the subject of each equation:
a 6x
+ 3y β 15 = 0 b 2x
β 5y β 9 = 0
c 3x
β 7y + 12 = 0 β GCSE MathematicsPrior knowledge check
Straight line graphs are used in mathematical
modelling. Economists use straight line graphs to model how the price and availability of a good affect the supply and demand.
βΒ ExerciseΒ 5HΒ Q95
|
[
0.024515662342309952,
0.05557006597518921,
0.0007674279040656984,
-0.04259038716554642,
-0.022908538579940796,
0.04945764318108559,
-0.05691401660442352,
-0.05434376373887062,
-0.10876592248678207,
0.01933881640434265,
0.0350445993244648,
-0.0353688970208168,
-0.008358057588338852,
0.00601972034201026,
-0.10732350498437881,
0.002365975407883525,
-0.07495336979627609,
0.0882340595126152,
-0.02727297879755497,
-0.06299695372581482,
0.0010254489025101066,
0.014944887720048428,
-0.04953176528215408,
-0.03830613195896149,
0.02649698406457901,
0.009985199198126793,
0.03708537295460701,
-0.011573150753974915,
0.006536273751407862,
-0.02595232054591179,
0.024383999407291412,
-0.027960576117038727,
0.10788077861070633,
0.04443933442234993,
-0.013313774019479752,
-0.07287658005952835,
0.11103235185146332,
0.05483574792742729,
0.02996188960969448,
-0.0877937451004982,
-0.08005405217409134,
0.05328163877129555,
-0.012919661588966846,
0.05493207648396492,
0.06601536273956299,
0.015419756062328815,
-0.044254861772060394,
-0.038245078176259995,
0.027852052822709084,
-0.0854509100317955,
0.022451404482126236,
-0.007485809735953808,
-0.047707699239254,
0.03691176325082779,
0.06837714463472366,
0.006677977275103331,
0.019052989780902863,
-0.0020702548790723085,
-0.04606962203979492,
0.05698990449309349,
0.023154115304350853,
-0.014029544778168201,
-0.03289353474974632,
0.03402809798717499,
-0.02594316005706787,
0.0315316841006279,
-0.00021415734954643995,
0.06524733453989029,
-0.04067728668451309,
0.0933765172958374,
-0.1575385183095932,
0.021940410137176514,
-0.033496588468551636,
-0.009358135052025318,
-0.013414433225989342,
0.02636897563934326,
0.022611254826188087,
0.04018283635377884,
-0.05806906148791313,
-0.1091044619679451,
-0.02291817031800747,
0.05513827130198479,
0.004730858840048313,
0.03440196067094803,
0.041806332767009735,
-0.034565143287181854,
0.03623059391975403,
0.07742723822593689,
-0.013943466357886791,
-0.019028961658477783,
0.04514259845018387,
0.017515387386083603,
-0.04747270047664642,
-0.02580154500901699,
0.054066259413957596,
0.028757257387042046,
-0.06770659238100052,
-0.0834055244922638,
0.004464380908757448,
0.06644953042268753,
-0.018406659364700317,
-0.017099084332585335,
-0.0371156707406044,
0.09179917722940445,
0.03901653736829758,
0.02418709173798561,
0.05894838646054268,
-0.013914366252720356,
0.007876367308199406,
-0.0720277950167656,
-0.05890914052724838,
0.02849048748612404,
0.0315999910235405,
-0.06729951500892639,
0.09768154472112656,
-0.038282789289951324,
0.021857919171452522,
-0.05350034683942795,
0.06991352885961533,
0.0207726638764143,
-0.004687414970248938,
-0.02487792819738388,
0.03547628968954086,
0.04223618283867836,
-0.05787378549575806,
0.06330801546573639,
-0.007612342946231365,
-0.006868393160402775,
0.008280878886580467,
-0.010600837878882885,
0.02845154143869877,
-0.021512525156140327,
-0.011142845265567303,
-0.006232015788555145,
-0.016666829586029053,
0.012175117619335651,
-0.008714618161320686,
0.05533025786280632,
-0.045442476868629456,
-0.05171816796064377,
0.09431871026754379,
0.12282302230596542,
0.0547158382833004,
-0.03369244933128357,
-0.041226305067539215,
-0.01646868884563446,
-0.05682587996125221,
0.014436410740017891,
-0.007655626628547907,
0.022594662383198738,
0.10177334398031235,
-0.03757340833544731,
0.03954280912876129,
0.07720016688108444,
-0.024528775364160538,
0.002495533088222146,
-0.00778214493766427,
0.01307659037411213,
-0.010656945407390594,
-0.025662284344434738,
-0.050477586686611176,
0.05216963589191437,
-0.03930447995662689,
0.05492508038878441,
0.0381874181330204,
-0.02965674176812172,
0.09317891299724579,
0.007312464527785778,
-0.05666134133934975,
0.05313311517238617,
0.07552728801965714,
-0.018829254433512688,
-0.005835942458361387,
0.1404510736465454,
-0.039394211024045944,
-0.03115716576576233,
0.13346266746520996,
0.006762255914509296,
-0.02101544849574566,
-0.04310256987810135,
-0.027189424261450768,
-0.03516183793544769,
0.026204312220215797,
-0.08460858464241028,
-0.009549584239721298,
0.03327464312314987,
0.010433772578835487,
0.13907380402088165,
-0.018370958045125008,
0.05671599134802818,
0.008845154196023941,
-0.024466820061206818,
-0.09621656686067581,
0.0528804212808609,
-0.1603577882051468,
-0.035268403589725494,
0.02226717583835125,
-0.020108584314584732,
-0.08070289343595505,
-0.11319419741630554,
0.03072918951511383,
-0.054168786853551865,
0.004743879660964012,
0.04842372238636017,
0.019131507724523544,
-0.0002709936525207013,
-0.03511859104037285,
-0.023572951555252075,
-0.0487380288541317,
0.0740835890173912,
0.02188301272690296,
-0.049399103969335556,
0.019435467198491096,
0.0030638144817203283,
0.0319012813270092,
-0.09023620188236237,
0.016075633466243744,
-0.09097817540168762,
-0.12346016615629196,
0.07031626999378204,
0.02038792334496975,
0.03544124215841293,
0.012028918601572514,
9.278662537143999e-33,
0.007954331114888191,
0.005085463635623455,
-0.05807686969637871,
-0.025599809363484383,
0.007689920254051685,
-0.01058698259294033,
0.0742715522646904,
-0.014107899740338326,
0.07325347512960434,
0.09668376296758652,
-0.008868718519806862,
0.030158042907714844,
-0.037228889763355255,
0.04606147110462189,
-0.016561849042773247,
-0.0559762604534626,
-0.012067887000739574,
-0.03414255380630493,
-0.006418891251087189,
-0.0148314218968153,
-0.014769205823540688,
-0.04898200184106827,
-0.02212367206811905,
0.043667566031217575,
0.02892088145017624,
-0.015522971749305725,
0.16089636087417603,
-0.09565705806016922,
-0.05332354083657265,
0.012957597151398659,
0.019228432327508926,
-0.07286500185728073,
-0.011798035353422165,
-0.016193248331546783,
-0.06665114313364029,
-0.01245421264320612,
0.0062114279717206955,
0.04517567530274391,
-0.04671629145741463,
-0.06425628811120987,
0.09718802571296692,
0.10079898685216904,
0.06926950067281723,
-0.10093367099761963,
-0.025708410888910294,
0.023104486986994743,
-0.015528984367847443,
0.007193182595074177,
-0.08194879442453384,
0.03593717887997627,
0.03676464408636093,
-0.03790849447250366,
0.08193214237689972,
-0.0004948019632138312,
0.0222829207777977,
-0.05477696284651756,
0.0020622864831238985,
-0.046551529318094254,
0.05820295959711075,
0.025319566950201988,
-0.067690409719944,
-0.037009142339229584,
-0.014245576225221157,
0.029411714524030685,
-0.025319186970591545,
-0.10274240374565125,
-0.055668096989393234,
-0.0304301455616951,
-0.04175088554620743,
-0.061136409640312195,
-0.034251801669597626,
0.08914671838283539,
0.023512646555900574,
-0.03943122923374176,
0.009246923960745335,
-0.03796909749507904,
0.017959747463464737,
-0.01137146633118391,
0.070005401968956,
-0.043376266956329346,
-0.047728944569826126,
-0.009650681167840958,
0.0479687862098217,
-0.009058733470737934,
0.039752423763275146,
-0.039538607001304626,
0.0357208289206028,
0.07134682685136795,
0.08768366277217865,
-0.02528301626443863,
-0.05963921546936035,
-0.03812647610902786,
-0.05215471237897873,
-0.05518774688243866,
0.063571497797966,
6.886111734800606e-32,
-0.1008506566286087,
0.04974609613418579,
0.03016859106719494,
0.04483357071876526,
0.019010894000530243,
0.0489802360534668,
0.009378471411764622,
0.03794575482606888,
-0.03427032753825188,
-0.0070785353891551495,
0.016587750986218452,
0.01849035732448101,
-0.0946556106209755,
0.05475214123725891,
0.037394169718027115,
-0.02469789981842041,
-0.008219928480684757,
0.051034871488809586,
0.004385926760733128,
-0.02840466983616352,
-0.036068908870220184,
0.04518922045826912,
-0.008480053395032883,
0.0762694776058197,
0.027097895741462708,
0.05445685610175133,
0.06884321570396423,
-0.015194974839687347,
0.016012227162718773,
-0.08484157919883728,
0.006499472074210644,
0.021691253408789635,
0.022796398028731346,
0.005236984696239233,
0.05377935618162155,
-0.006041174288839102,
-0.07392355054616928,
0.06191495433449745,
-0.039376191794872284,
0.03231145441532135,
-0.00010635595390340313,
-0.048147473484277725,
0.0415164940059185,
-0.057596806436777115,
0.03342089056968689,
0.0378749743103981,
-0.08879755437374115,
-0.05192781239748001,
-0.06559915840625763,
-0.008249299600720406,
0.03260447084903717,
0.025408528745174408,
0.03435434401035309,
-0.020316055044531822,
-0.03654193505644798,
-0.059466857463121414,
0.042043980211019516,
-0.013372756540775299,
-0.08397312462329865,
-0.005979740060865879,
-0.0023401672951877117,
0.08552125841379166,
-0.09506744891405106,
-0.03503115847706795
] |
90
Chapter 5
y
x O(x2, y2)
(x1, y1)x2 β x 1y2 β y 15.1 y = mx + c
You can find the gradient of a straight line joining two points
by considering the vertical distance and the horizontal distance between the points.
β The gradient m of a line joining the point with coordinates
( x 1 , y 1 ) to the point with coordinates ( x 2 , y 2 )
can be calculated using the f
ormula m = y2 β y1 ______ x2 β x1
a β (β5) ________ 4 β 2 = β1
So a +
5 ______ 2 = β1
a + 5 = β 2
a = β7Example 2
The line joining (2, β5) to (4, a) has gradient β1. Work out the value of a.
Use m = y2 β y1 ______ x2 β x1 . Here m = β1, (x1, y1) = (2, β5)
and (x2, y2) = (4, a).Example 1
Work out the gradient of the line joining (β2, 7) and (4, 5)
y
x O(β2, 7)
(4, 5)
m = 5 β 7 _______ 4 β (β2) = β 2 __ 6 = β 1 __ 3 Use m = y2 β y1 ______ x2 β x1 . Here (x1, y1) = (β2, 7) and
(x2, y2) = (4, 5)
1 Work out the gradients of the lines joining these pairs of points:
a (4, 2), (6, 3) b (β1, 3), (5, 4) c (β4, 5), (1, 2)
d (2, β3), (6, 5) e (β3, 4), (7,
β6) f (β
12, 3), (β2, 8)
g (β2,
β4), (10, 2) h ( 1 _ 2 , 2), ( 3 _ 4 , 4) i ( 1 _ 4 , 1 _ 2 ), ( 1 _ 2 , 2 _ 3 )
j (β2.4, 9.6), (0, 0) k (1.3, β2.2), (8.8,
β4.7) l (0, 5a), (10
a, 0)
m (3b
, β2b), (7b, 2b) n ( p,
p2), (q, q2)Exercise 5A Explore the gradient
fo
rmula using GeoGebra.Online
|
[
-0.01725941151380539,
-0.0033452033530920744,
0.028358004987239838,
0.00020366208627820015,
-0.0051992908120155334,
0.08801115304231644,
0.021469373255968094,
-0.03211066499352455,
-0.0009195214370265603,
0.03231723606586456,
0.0053368001244962215,
-0.010094579309225082,
0.08013003319501877,
0.030854662880301476,
-0.07312855869531631,
-0.021006044000387192,
-0.032479315996170044,
0.02287384867668152,
-0.05513095110654831,
-0.035003580152988434,
-0.0025821903254836798,
-0.01637071557343006,
-0.09877581894397736,
-0.010712381452322006,
0.006488148123025894,
0.04899671673774719,
-0.04484780132770538,
-0.013296783901751041,
0.0027456197421997786,
-0.05874849855899811,
0.03971012309193611,
-0.040478430688381195,
0.03670957684516907,
0.04880743473768234,
0.052539579570293427,
0.02624373883008957,
0.07114172726869583,
0.022585365921258926,
0.0029709695372730494,
-0.003155570477247238,
-0.08243684470653534,
0.009606365114450455,
-0.00023770108236931264,
0.0027352692559361458,
-0.04245533421635628,
0.021913906559348106,
0.0166401918977499,
0.014384171925485134,
0.0013018136378377676,
0.0135755380615592,
0.0832776352763176,
-0.0600474588572979,
-0.11003987491130829,
-0.01144260074943304,
-0.02196006290614605,
0.04861878603696823,
-0.0014693388948217034,
-0.07640669494867325,
0.047124963253736496,
0.0772463008761406,
-0.011448071338236332,
0.021545521914958954,
-0.005571629386395216,
0.06525574624538422,
-0.06931786239147186,
-0.0577671192586422,
0.05085127800703049,
-0.06623861938714981,
-0.04198634624481201,
0.058949150145053864,
-0.11682716757059097,
-0.03102508746087551,
0.0294193048030138,
-0.04348469525575638,
-0.03165562078356743,
0.023490697145462036,
0.03898884356021881,
-0.036130741238594055,
-0.04798157513141632,
-0.12817129492759705,
-0.0009294934570789337,
0.05686476454138756,
-0.04527997225522995,
0.03729316219687462,
-0.028257200494408607,
-0.020553553476929665,
-0.016994407400488853,
0.085089311003685,
0.014293225482106209,
-0.011177588254213333,
0.0709589496254921,
0.02244235761463642,
-0.1159120723605156,
0.030431285500526428,
0.027636313810944557,
0.04942752793431282,
0.0009842412546277046,
-0.06305156648159027,
0.028696373105049133,
0.07730285823345184,
-0.020605456084012985,
0.06846573948860168,
-0.06798230111598969,
0.0033069984056055546,
0.09108921885490417,
0.08653748035430908,
0.07623977959156036,
-0.01948884315788746,
0.08040644228458405,
-0.02812378667294979,
-0.09253088384866714,
-0.02140915021300316,
0.00687052309513092,
0.013439084403216839,
0.08117364346981049,
0.02177254483103752,
0.07511758804321289,
-0.03647797927260399,
0.09026086330413818,
-0.04991183429956436,
0.0012182972859591246,
-0.12981252372264862,
-0.005220666527748108,
0.0146377794444561,
-0.04555141180753708,
0.005056579131633043,
0.005388204008340836,
-0.055150214582681656,
0.01001674309372902,
-0.020519623532891273,
-0.051620565354824066,
-0.04653065279126167,
-0.02480553276836872,
-0.010103123262524605,
-0.025037117302417755,
0.020463738590478897,
-0.0956520140171051,
0.02537517249584198,
-0.05811242386698723,
-0.04875911772251129,
0.058843158185482025,
0.034165214747190475,
0.061687834560871124,
0.00048607971984893084,
-0.05063805729150772,
-0.010525288060307503,
-0.0008296385640278459,
0.06244994327425957,
0.004659153986722231,
0.051751188933849335,
0.049302730709314346,
-0.05697285011410713,
-0.015492626465857029,
-0.006064575631171465,
-0.01915055513381958,
-0.06436465680599213,
-0.009142591618001461,
0.014931801706552505,
0.017737215384840965,
-0.04594741761684418,
-0.07665970921516418,
0.007173970807343721,
-0.0814111977815628,
0.035194557160139084,
0.031123431399464607,
0.0015187477692961693,
0.04567409306764603,
0.09334749728441238,
0.03502470999956131,
0.09618700295686722,
0.05117536708712578,
-0.0031768789049237967,
-0.1023637056350708,
0.030842110514640808,
-0.07509186863899231,
-0.0070144482888281345,
0.07140782475471497,
0.05057590827345848,
0.0003525081556290388,
-0.030010385438799858,
0.00937341433018446,
-0.03617478534579277,
-0.01753932610154152,
-0.026812709867954254,
-0.02924385480582714,
-0.015315096825361252,
0.004075601696968079,
0.018546219915151596,
-0.08909332752227783,
0.03683600574731827,
0.04474906250834465,
-0.06123987212777138,
-0.03852391988039017,
0.04010351747274399,
-0.11284234374761581,
0.002023152308538556,
0.061011236160993576,
-0.044196367263793945,
0.008086911402642727,
-0.03880145773291588,
0.033167775720357895,
0.013500499539077282,
-0.032187141478061676,
0.023395219817757607,
0.016760682687163353,
-0.05193537473678589,
-0.0006865279865451157,
-0.07976164668798447,
-0.109189972281456,
0.049983978271484375,
0.052936457097530365,
0.010125662200152874,
-0.05706052482128143,
0.049009647220373154,
0.0024124227929860353,
-0.07639241963624954,
0.010615939274430275,
0.019139815121889114,
-0.05209007486701012,
-0.013867142610251904,
-0.02896863780915737,
0.059246405959129333,
-0.006906083784997463,
-7.198495244301401e-33,
-0.03287258371710777,
-0.018052583560347557,
0.005269412416964769,
-0.06898506730794907,
-0.0035701121669262648,
-0.06928517669439316,
0.050304118543863297,
-0.018815094605088234,
-0.006990504451096058,
0.20099090039730072,
-0.02018597349524498,
0.00108105328399688,
0.00006518482405226678,
0.019652169197797775,
0.03920203447341919,
-0.047508951276540756,
0.016537779942154884,
0.04679342731833458,
-0.001440729247406125,
-0.04781217873096466,
-0.0012716040946543217,
-0.04100159928202629,
-0.026118118315935135,
-0.024905536323785782,
0.03127650171518326,
-0.03863608092069626,
0.048908475786447525,
-0.009364602155983448,
-0.02579171024262905,
0.04380160570144653,
-0.03827127814292908,
-0.06700283288955688,
0.014641928486526012,
0.050967756658792496,
-0.026090426370501518,
0.038082778453826904,
-0.08316443115472794,
0.014802565798163414,
-0.00010209889296675101,
-0.12238462269306183,
0.022901099175214767,
0.09940024465322495,
0.12453727424144745,
-0.054034408181905746,
0.003492729738354683,
0.08274681121110916,
-0.017751997336745262,
-0.0705195739865303,
-0.002189805032685399,
0.025233741849660873,
-0.01877688057720661,
-0.05139591172337532,
-0.005809290334582329,
0.03748196363449097,
-0.053833670914173126,
-0.05537354201078415,
-0.08315461128950119,
-0.006248533260077238,
0.021114574745297432,
-0.03664065897464752,
-0.01876017265021801,
0.020921457558870316,
-0.042865291237831116,
0.059008341282606125,
0.030980300158262253,
0.02096143178641796,
-0.07304911315441132,
0.013637772761285305,
0.011152010411024094,
-0.019490284845232964,
-0.10202606767416,
0.06256521493196487,
0.0246672835201025,
0.022118045017123222,
-0.06425543129444122,
-0.0788612887263298,
0.0496368445456028,
-0.006957066245377064,
0.06375196576118469,
-0.006284326780587435,
-0.009722459129989147,
0.057976994663476944,
0.040537044405937195,
0.0015258266357704997,
0.03807086497545242,
0.03554176166653633,
0.01756521686911583,
0.03645716980099678,
0.06399504840373993,
-0.003269483568146825,
-0.10322372615337372,
0.04624304547905922,
-0.03800496459007263,
-0.07367246598005295,
0.030618799850344658,
8.171814558484237e-32,
-0.15847645699977875,
-0.015046617016196251,
-0.0013535915641114116,
0.044391825795173645,
0.05344010517001152,
0.09318965673446655,
-0.016502631828188896,
-0.013362592086195946,
0.016011444851756096,
-0.06283984333276749,
-0.029939496889710426,
0.011172694154083729,
-0.020612429827451706,
0.08506201207637787,
0.03942996636033058,
0.03277210518717766,
-0.00836851168423891,
-0.035994790494441986,
-0.041720300912857056,
-0.00883703213185072,
-0.019724519923329353,
0.011011680588126183,
-0.08208242803812027,
0.0005771709838882089,
0.009419315494596958,
-0.025275496765971184,
-0.012783280573785305,
0.0353308767080307,
0.0482599139213562,
-0.06579262018203735,
0.07187971472740173,
-0.07972445338964462,
0.002251390600576997,
0.09007725864648819,
0.06229833886027336,
0.019757824018597603,
-0.056405872106552124,
0.05136914923787117,
0.012996471486985683,
-0.006067371927201748,
-0.010804684832692146,
-0.036645274609327316,
0.0025898825842887163,
-0.050846438854932785,
0.02578093856573105,
-0.0019111076835542917,
-0.06679516285657883,
-0.08465499430894852,
-0.062373947352170944,
0.0038483371026813984,
0.11613749712705612,
0.05601492524147034,
0.08439148962497711,
0.10944615304470062,
-0.0926763191819191,
-0.11990843713283539,
0.13391079008579254,
-0.05419256165623665,
0.02238485962152481,
0.04869760572910309,
-0.042101722210645676,
0.05359265208244324,
-0.07174032926559448,
-0.04419231414794922
] |
91Straight line graphs
2 The line joining (3, β5) to (6,
a) has a gradient 4. Work out the value of a.
3 The line joining (5, b) to (8, 3) has gr
adient β3. Work out the value of b.
4 The line joining (c, 4) to (7, 6) has gr
adient 3 _ 4 . Work out the value of c.
5 The line joining (β1, 2
d ) to (1, 4) has gradient β 1 _ 4 . Work out the value of d.
6 The line joining (β3,
β2) to (2e, 5) has gradient 2. Work out the value of e.
7 The line joining (7, 2) to ( f, 3f ) has gradient 4. Wor
k out the value of f.
8 The line joining (3, β4) to (
βg, 2g) has gradient β3. Work out the value of g.
9 Show that the points
A(2, 3), B(4, 4) and
C(10,7) can be joined by a straight line.
10 Show that the points
A(β2a, 5a), B(0, 4a)
and points C(6a, a) are collinear. (3 marks)
β The equation of a straight line can be writt
en in the form
y = mx + c, where m is the gradient and c is the y-intercept.
β The equation of a straight line can also be writt
en in the
form ax + by + c = 0, where a, b and c are integers.P
E/P Poi nts are collinear if they all lie on
the same straight line.NotationFind the gradient of the line joining the points A
and B and the line joining the points A and C .Problem-solving
y
x Oy = mx + c
cm
1
a Gradient = β3 a nd y-intercept = (0, 2).
b y =
4 __ 3 x + 5 __ 3
Grad
ient = 4 __ 3 and y -intercept = (0, 5 __ 3 ).Example 3
Write down the gradient and y-intercept of these lines:
a y = β3
x + 2 b 4x β
3y + 5 = 0
Use f ractions rather than decimals
in coordinate geometry questions.Watch outRearrange the equation into the form y = mx + c.
From this m = 4 _ 3 and c = 5 _ 3 Compare y = β3x + 2 with y = mx + c.
From this, m = β3 and c = 2.
|
[
0.028725745156407356,
0.03366002440452576,
0.0209120512008667,
-0.06412093341350555,
-0.03791247680783272,
0.03455302119255066,
-0.01980348862707615,
-0.07750751078128815,
-0.08368227630853653,
0.023953501135110855,
0.01275207195430994,
-0.03535762056708336,
-0.02082219533622265,
-0.013173596002161503,
-0.07926227897405624,
0.009692234918475151,
-0.02318441867828369,
0.06149446591734886,
-0.12027380615472794,
-0.038262005895376205,
-0.03926714137196541,
-0.043625976890325546,
-0.09027866274118423,
-0.019722163677215576,
0.08273126929998398,
-0.009328853338956833,
0.018371082842350006,
0.012924172915518284,
0.040014710277318954,
-0.07336835563182831,
0.06265068054199219,
-0.022165430709719658,
0.0559401661157608,
0.02749253436923027,
0.04245690256357193,
-0.017481226474046707,
0.07471994310617447,
0.09328348189592361,
-0.005451641511172056,
-0.038785841315984726,
-0.05544716864824295,
-0.018160728737711906,
-0.05720539018511772,
0.005848378874361515,
-0.0020723254419863224,
0.03829880431294441,
-0.011685275472700596,
-0.011686091311275959,
0.0029958824161440134,
0.02292974293231964,
0.01681428775191307,
-0.0052406364120543,
-0.07500328123569489,
-0.019351335242390633,
0.006646143738180399,
0.0023866123519837856,
0.053629692643880844,
-0.04373347386717796,
0.013456995598971844,
0.04697349667549133,
0.03890303522348404,
0.05065923556685448,
-0.05799168348312378,
0.03799733519554138,
-0.027376336976885796,
0.08066597580909729,
-0.007718899752944708,
-0.03846507519483566,
-0.06107703596353531,
0.046565622091293335,
-0.12285402417182922,
0.008525070734322071,
-0.012790157459676266,
-0.12349630147218704,
-0.036760393530130386,
0.01623208075761795,
0.017462152987718582,
-0.021809367462992668,
-0.06448815762996674,
-0.11643651872873306,
-0.05358780547976494,
0.09902278333902359,
0.008929690346121788,
0.10043228417634964,
0.015079223550856113,
-0.00913329515606165,
0.012926317751407623,
0.09220252931118011,
-0.013893899507820606,
0.001770099624991417,
0.029447034001350403,
0.06978403031826019,
-0.05944089591503143,
0.006261141505092382,
0.01825954020023346,
-0.009226275607943535,
-0.027619419619441032,
-0.05384621024131775,
0.009976156055927277,
0.09876440465450287,
0.013833642937242985,
0.0191135685890913,
-0.11426035314798355,
0.06671672314405441,
0.01195453479886055,
0.01948269084095955,
0.04272424429655075,
-0.024313392117619514,
-0.005271310918033123,
-0.04757649078965187,
-0.003013191744685173,
-0.02715630643069744,
0.0302740428596735,
-0.03420122340321541,
0.06748321652412415,
-0.049964968115091324,
-0.02543243207037449,
-0.012225558049976826,
0.08673904091119766,
-0.041572991758584976,
0.0504705011844635,
-0.08224185556173325,
0.06945569813251495,
-0.031076928600668907,
-0.08453961461782455,
-0.01716979220509529,
-0.015332868322730064,
0.032412443310022354,
0.018689114600419998,
-0.018151499330997467,
0.03085312992334366,
-0.05571379140019417,
-0.006748579442501068,
-0.04969284310936928,
-0.08667577058076859,
-0.051213618367910385,
-0.04883338510990143,
0.06594701111316681,
-0.030556006357073784,
-0.06780711561441422,
0.02642819844186306,
0.0843837633728981,
0.03837452828884125,
-0.002268117154017091,
-0.03656025975942612,
-0.03158088028430939,
-0.06384836882352829,
-0.02220766991376877,
0.009594187140464783,
0.05653009191155434,
0.033294446766376495,
-0.026833435520529747,
0.06162724271416664,
0.055086519569158554,
-0.045177292078733444,
-0.01648394949734211,
-0.03161457180976868,
0.007852084003388882,
-0.029592519626021385,
-0.03194078058004379,
-0.059351660311222076,
0.09086635708808899,
-0.02121410146355629,
0.06641387194395065,
0.06272367388010025,
0.04527902975678444,
0.09235728532075882,
-0.00866993609815836,
0.026145942509174347,
-0.005193023011088371,
0.04630396142601967,
0.007155325263738632,
-0.045588068664073944,
0.1459764540195465,
-0.01325797475874424,
-0.03687560185790062,
0.0909789428114891,
0.015595235861837864,
-0.0011616848642006516,
-0.05116969719529152,
0.0685558095574379,
-0.02370358631014824,
-0.00019440478354226798,
-0.031114233657717705,
-0.013174616731703281,
-0.023767590522766113,
0.018147684633731842,
0.10845210403203964,
-0.08569823950529099,
0.0441700704395771,
0.04331605136394501,
-0.03305387124419212,
-0.0660429298877716,
0.05725012719631195,
-0.1352936029434204,
-0.032802481204271317,
0.06183647736907005,
0.00328175351023674,
-0.10589371621608734,
-0.07146687060594559,
0.09429092705249786,
0.030598841607570648,
0.028242329135537148,
0.04706883430480957,
-0.011527798138558865,
0.007672043051570654,
0.0014304679352790117,
-0.04129081591963768,
-0.058394886553287506,
0.030163444578647614,
0.05157538503408432,
0.017397327348589897,
-0.04857313260436058,
0.01846870221197605,
0.04959683120250702,
-0.08459336310625076,
0.02773221768438816,
-0.06081831082701683,
-0.10520947724580765,
0.029755529016256332,
-0.021287959069013596,
0.025226643308997154,
0.013446444645524025,
3.0013559764097086e-34,
-0.03812748193740845,
0.06065317988395691,
-0.0331522673368454,
-0.0325697660446167,
0.015397106297314167,
0.002960074692964554,
0.08728672564029694,
-0.028983546420931816,
0.028208758682012558,
0.1360044926404953,
0.011845680885016918,
-0.003604752477258444,
-0.0610223151743412,
0.035707540810108185,
0.026836469769477844,
-0.0653521716594696,
-0.01547167543321848,
-0.009294050745666027,
-0.02771640010178089,
-0.05224933102726936,
0.02239750698208809,
-0.04051724076271057,
-0.09789150953292847,
0.02668525092303753,
0.09440837800502777,
-0.0006862437003292143,
0.05584869906306267,
-0.10180367529392242,
-0.030766058713197708,
-0.03284787759184837,
0.010947637259960175,
-0.024000050500035286,
0.025432974100112915,
0.02605349011719227,
-0.06164420023560524,
-0.009370685555040836,
0.04332973062992096,
-0.031926799565553665,
-0.01699814759194851,
-0.045173704624176025,
0.06265070289373398,
0.14654840528964996,
0.016886243596673012,
-0.05365941300988197,
-0.06548243761062622,
0.036876726895570755,
0.026409877464175224,
-0.01129649206995964,
-0.03325530141592026,
0.05151712894439697,
0.027293914929032326,
-0.037796009331941605,
0.04893777519464493,
0.05088810995221138,
0.044278185814619064,
-0.03422415629029274,
0.029900526627898216,
-0.00043933058623224497,
0.04645409435033798,
0.03891102597117424,
-0.08162357658147812,
-0.1026773527264595,
-0.0076517886482179165,
0.059629082679748535,
-0.0023959060199558735,
-0.03514181450009346,
-0.028974078595638275,
-0.012786678038537502,
0.017205774784088135,
-0.03887607157230377,
0.013199597597122192,
0.08129525929689407,
0.011345630511641502,
-0.05689641088247299,
0.008827265352010727,
-0.06867073476314545,
0.013333898037672043,
0.004913826938718557,
0.06731590628623962,
-0.07417979091405869,
-0.04719115048646927,
-0.04420087859034538,
0.08306919783353806,
0.035040829330682755,
0.037779223173856735,
-0.01836157776415348,
0.03705313429236412,
0.06359993666410446,
0.11829305440187454,
0.05273128300905228,
-0.09849663078784943,
0.010307453572750092,
-0.07929801195859909,
-0.06768310070037842,
0.06231747940182686,
7.554576580799664e-32,
-0.03828892856836319,
0.03369961678981781,
-0.044931355863809586,
0.04378712549805641,
0.01684800535440445,
0.047352951020002365,
0.02189624309539795,
-0.010588780976831913,
-0.07317567616701126,
0.005660394672304392,
0.03301111236214638,
0.025854431092739105,
-0.10498877614736557,
0.05645810812711716,
0.03847747668623924,
-0.010373626835644245,
-0.04939596354961395,
-0.008015209808945656,
-0.016579678282141685,
-0.0716734454035759,
-0.06666646152734756,
0.02706189453601837,
0.025041488930583,
0.07140521705150604,
0.037185188382864,
0.0005245743086561561,
0.03332531452178955,
-0.014379452913999557,
0.030976973474025726,
-0.06310250610113144,
-0.0005698289023712277,
0.013974359259009361,
0.023357482627034187,
0.00004762494791066274,
0.04481852427124977,
0.04466832056641579,
-0.0999884307384491,
0.05534820631146431,
-0.019889790564775467,
0.02451646514236927,
0.05383927747607231,
-0.04937046766281128,
0.044463805854320526,
-0.07745593786239624,
0.003560652257874608,
0.008949283510446548,
0.0028188559226691723,
-0.032753828912973404,
-0.05756852403283119,
-0.009071222506463528,
0.03486994653940201,
0.08314740657806396,
0.03634997084736824,
0.012779681012034416,
-0.051185160875320435,
-0.11569973081350327,
0.09200131893157959,
0.01999444141983986,
-0.04484980180859566,
-0.06181854009628296,
0.003546077525243163,
0.03113127313554287,
-0.11321833729743958,
-0.03239581733942032
] |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.